The Merseysiders twice fell behind at Craven Cottage on Wednesday, with Kolo Toure slicing past his own goalkeeper and then Kieran Richardson making the most of Martin Skrtel's inadvertent deflection to score from close range.
On both occasions, though, Liverpool bounced back. First, Gerrard's exquisite pass was received by Daniel Sturridge - who kept his cool with a neat finish - before Philippe Coutinho cut inside to strike home from 20 yards.
It looked like Liverpool would be frustrated in their efforts to find a winner until Sturridge was chopped down by Sascha Riether in the last minute, and Gerrard held his nerve to secure three vital points from the spot.
The result leaves Liverpool just four points adrift of Chelsea at the top of the Premier League, and Rodgers waxed lyrical about the performance of his captain.
"Steven is peerless in many aspects of football," he said. "His performance again was (brilliant).
"His pass for the first goal, that penetration he has that he's shown all his life and then the composure at the end.
"It's no coincidence, he studies the goalkeepers and what side they dive. The professionalism he's showing at this stage of his career is still there.
"There's no more ideal person than Steven Gerrard when you have a penalty in the last couple of minutes in games, he stuck it away brilliantly."
Rodgers once more refused to be drawn on his side's title hopes and said his main focus is on ensuring that Liverpool continue to improve.
He added: "There's still a long time to go, there's 12 games. Our concentration since I've been here has always been the next game, getting on the training ground and improving aspects of our game.
"Here you've seen the development of the team in terms of character.
"After the game at the weekend (a 5-1 win over Arsenal) we were outstanding. We had to show different qualities tonight, one of those being character and the togetherness there is in this group."
