Robin van Persie had United's best chances on his return to Arsenal, but failed to make the most of a one-on-one in the early stages and then saw a late header saved by Wojciech Szczesny.
The result, combined with Liverpool's win, further damaged United's ambition as the Red Devils are now 11 points behind the Champions League places.
"I see it as a positive result, it's a hard place to come and they've had a great season up to now,'' United manager David Moyes said, having failed again in his bid to secure a first away win against a top-four club as a manager. ``I'd have liked to have won tonight, but there were a lot of good things to come out of it.''
Manchester City is three points behind Chelsea in third place after its dress rehearsal of the League Cup final against Sunderland was postponed because of bad weather. Everton's game against Crystal Palace was also called off because of stormy conditions in northwest England.
Liverpool, which has not lost in the league since late December, had to wait until the last minute to earn a dramatic win at Craven Cottage.
"We're going to keep pushing. We've shown enough times we're a very good side,'' Rodgers said.
"We're chasing, other teams are still ahead of us. We need some of them to drop points. We know we still have Man City and Chelsea to play at home. We'll take it game by game. I'm enjoying as a coach watching the team improve.''
Kolo Toure's own goal gave Fulham the lead before Daniel Sturridge leveled following Gerrard's superb pass. Kieran Richardson then made the most of poor defending by Martin Skrtel but Philippe Coutinho equalised from 20 yards before Gerrard's penalty.
Elsewhere, Stoke drew 1-1 with Swansea.
Emmanuel Adebayor scored twice as Tottenham thrashed Newcastle at St. James' Park to maintain its push for a Champions League spot. It was Newcastle's fourth straight home defeat for the first time in the Premier League era. Brazil midfielder Paulinho and Nacer Chadli completed the rout for Spurs.
Tottenham stays fifth, three points behind Liverpool.
"Adebayor hasn't really surprised me because I've seen him be fantastic for many clubs he's played for,'' Spurs manager Tim Sherwood said. ``I'm delighted he's in such great form. His concentration levels and the way he trains is magnificent, he's a good example to all young players, especially at our club.''