Lot succeeded where Avraham failed.
Avraham negotiated with God regarding the destruction of Sedom, but only managed to settle on the agreement that God will spare the city if there would be ten righteous people within it, which proved not to be the case.
However, Lot requested to live in the small city of Zo'ar and thus preventing its destruction, and God agreed: 'See, I have accepted thee concerning this thing also, that I will not overthrow the city of which you have spoken.' Lot managed to persuade God to save the city on behalf of only one righteous person.
How can one explain that Lot was more successful in his endeavor than Avraham?
Perhaps we can answer this question by examining the cities in which Avraham and Lot lived. Avraham fought to save Sedom out of beliefs of righteousness and justice, but looking at the situation as an outsider.
However, Lot actually lived within the corrupt society of Sedom yet still followed Avraham's path of righteousness and kindness. This granted him an additional merit when he requested that one of Sedom's cities be spared.
As proven by the births of his grandchildren, Amon and Moav, Lot was distinctly less righteous than Avraham in many areas. Nevertheless, his dedication to his own path even while living in the corrupt city of Sedom itself raised Lot to a high level - granting him the ability to save an entire city on his own merit, specifically because he was living within that society.