The Reds trailed twice at Craven Cottage against the rock-bottom Cottagers, but managed to steal all three points courtesy of a Steven Gerrard penalty-kick in injury time.
The result came on the back of a stunning 5-1 demolition of Arsenal at Anfield over the weekend, and moved the Merseysiders to within four points of leaders Chelsea after they could only draw at home with West Brom earlier this week.
Rodgers says that his side are reliant upon their rivals slipping up between now and the end of the season, but that they have a chance of lifting the league trophy later this year:
“We knew this was important,” he told Sky Sports.
“I said to the players these are the games where you arrive in the Champions League – games like this here. You’re playing a team who are fighting, trying to stay in the division.
“We’ve come off the back of a brilliant performance at the weekend and tonight we’ve again got another three points in a different way.
“So it really shows that this team has improved in every facet of the game.
“Seeing Steven (Gerrard) take off his shirt and show off his muscles was great.
“We’re just going to keep pushing and I think we’ve shown enough times that we’re a very good side. We’re chasing, but we need some of them (the teams above) to drop points before anything else can happen.
“We know we have Chelsea and Manchester City to play at home which we’ll look forward to because we’ve been outstanding at home but we’ve just got to take it game by game.
“I’m enjoying watching the team improve and develop as a coach.”
Daniel Sturridge and Philippe Coutinho twice pulled Liverpool back from behind after a Kolo Toure own goal and Kieran Richardson effort had put Fulham in the driving seat.
But a mistimed challenge from Sascha Riether on Sturridge gave Gerrard the chance to convert from the spot late on.