Praise be to Allaah.

Firstly: 

Bribery is a major sin, because of the report narrated by Ahmad (6791) and Abu Dawood (3580) from ‘Abd-Allaah ibn ‘Amr (may Allaah be pleased with him) who said: The Messenger of Allaah (peace and blessings of Allaah be upon him) cursed the one who gives a bribe and the one who takes it. Classed as saheeh by al-Albaani in Irwa’ al-Ghaleel (2621). 

So if you can get the work done without paying a bribe, then it is haraam for you to pay it. 

Secondly: 

If a person who has a right cannot get his right except by paying a bribe, then the scholars (may Allaah have mercy on him) have stated that it is permissible for him to give the bribe in that case, and the prohibition applies to the one who takes it and not the one who gives it. They quoted as evidence for that the report narrated by Ahmad (10739) from ‘Umar ibn al-Khattaab, according to which the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “One of them asks me for something and I give it to him, and he goes out with it, carrying it under his arm, and it is nothing but fire for him.” ‘Umar said: O Messenger of Allaah, why do you give it to them?” He said: “They insist on asking me, and Allaah insists that I should not be stingy.” Classed as saheeh by al-Albaani in Saheeh al-Targheeb, 844. 

The Prophet (peace and blessings of Allaah be upon him) gave money to these people even though it was haraam for them, so as to ward off any accusation of stinginess from himself. 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: If he gives him a gift so that he will refrain from wronging him or so that he will give him his rights, this gift is haraam for the one who takes it, but it is permissible for the giver to give it to him, as the Prophet (peace and blessings of Allaah be upon him) said: “I give a gift to one of them…” End quote from al-Fataawa al-Kubra (4/174). 

He also said: 

The scholars said: it is permissible to give a bribe to an official so as to ward off mistreatment, not to make him withhold a right, but his taking the bribe is haraam. For example, if a man gives something to a poet or someone else so that he will not tell lies about him in verse or otherwise, or so that he will not say something about his honour that it is haraam for him to say, it is permissible for him to give that to him, but what he takes so that he will not wrong him is haraam for him, because he should not wrong him anyway. 

Everyone who takes money so that he will not tell lies about people or so that he will not mistreat them, all of that is haraam earnings, because wronging and lying are haraam anyway, so he should refrain from them without any compensation that he takes from the one who has been wronged. If he does not refrain from doing that unless he is paid, then this is haraam earnings. End quote. 

Majmoo’ al-Fataawa (20/252) 

He also said (31/278): 

The scholars said: The one who gives a gift to the one who is in authority so that he will do something for him that is not permissible, it is haraam for both the one who gives it and the one who takes it. This is the kind of bribe of which the Prophet (peace and blessings of Allaah be upon him) said: “Allaah has cursed the one who gives a bribe and the one who takes it.” 

But if he gives him a gift so that he will refrain from wronging him or so that he will give him a right that is his due, then this gift is haraam for the one who takes it but it is permissible for the giver to give it to him. The Prophet (peace and blessings of Allaah be upon him) used to say: “I give one of them something and he goes out with it under his arm, but it is fire.” It was said: O Messenger of Allaah, why do you give to them? He said: “They insist on asking me, and Allaah insists that I should not be stingy.” 

An example of that is giving to one who wrongs the people; it is permissible for the giver to give to him, but it is haraam for him to take it. 

With regard to giving in the case of intercession, such as when a man intercedes with the authorities to ask that a wrong be stopped or so that he may be given his rights or be appointed to a position that he deserves, so that he will be enrolled in the armed forces – and he deserves that – or so that he will be given money that comes from a waqf for the poor or fuqaha’ or students of Qur’aan or pilgrims and so on – and he is entitled to that – and other types of intercession which help people to do obligatory actions or to avoid haraam actions, in this case too it is not permissible to accept the gift, but it is permissible for the giver to give whatever will enable him to take his rights or ward off mistreatment. This is what is narrated from the salaf and the major imams. End quote.  

Taqiy al-Deen al-Subki (may Allaah have mercy on him) said: What is meant by the bribe that we have mentioned is what is given so as to ward off a right or obtain something falsely. If it is given in order to attain one’s right, then the prohibition applies to the one who takes it. But if the one who gives it cannot attain his rights except by this method, then it is permissible, but if he can attain his rights without it, then it is not permissible.  

Fataawa al-Subki, 1/204 

Al-Suyooti said in al-Ashbaah wa’l-Nazaa’ir (p. 150): 

The twenty-seventh principle: “What is haraam to take is also haraam to give”, such as riba, the wages of a prostitute, the fee of a fortuneteller, bribes, and payments to wailers and musicians. 

Some kinds are exempted, such as bribes paid to rulers in order to attain one's rights and free prisoners, and giving something to one whose slander in verse one fears. End quote. 

Al-Hamawi (Hanafi) said in Ghamz ‘Ayoon al-Basaa’ir: 

The fourteenth principle:  

“What is haraam to take is also haraam to give”, such as riba, the wages of a prostitute, the fee of a fortuneteller, bribes, and payments to wailers and musicians,” except in certain cases: 

1 – A bribe when one fears for one's wealth or oneself. 

This applies to the giver, but for the one to whom it is given it is haraam. End quote. 

It says in al-Mawsoo’ah al-Fiqhiyyah: 

In al-Ashbaah by Ibn Nujaym (Hanafi) and in al-Manthoor by al-Zarkashi (Shaafa’i) it says: “What is haraam to take is also haraam to give”, such as riba, the wages of a prostitute, the fee of a fortuneteller and bribes given to a judge so thate will pass an unlawful judgement, except in certain cases, such as a bribe given when a person fears for himself or his wealth, or in order to free a prisoner, or for one whose slander in verse one fears. 

Dr. Wahbah al-Zuhayli said: If a bribe is the only means of attaining a lawful purpose, then it is permissible to give it in cases of necessity, but it is haraam for the one who takes it. End quote. 

Conclusion: 

It is permissible for you to give a bribe and it is haraam for the official who takes it, but there are two conditions: 

1-     You should be giving it in order to take your rights or to ward off harm from yourself. But if you are giving it in order to take something to which you are not entitled, then it is haraam and is a major sin.

2-     There should be no other means of obtaining your rights or warding off harm from yourself except this bribe. 

And Allaah knows best.