Gerrard scored twice from the penalty spot at Old Trafford on Sunday and missed the chance to become the first Premier League player to score a hat-trick of spot-kicks in the same game before Luis Suarez added a third late on.
United finished with 10 men after Nemanja Vidic received his second yellow card for bringing down Daniel Sturridge during the third penalty incident and the result leaves David Moyes' men seventh.
In contrast, Liverpool are second, four points off leaders Chelsea and with a game in hand on Jose Mourinho's side.
Gerrard knows there is plenty of work to do for Brendan Rodgers' men, but warned the other sides at the top of the table that Liverpool will battle until the end of the season.
"We've shown we're genuine contenders and we'll fight to the end," he told Sky Sports.
"We believe it but the easy thing is to talk the talk. The rest of the people fighting with us have to believe we're going for it."
The win was Liverpool's first at Old Trafford in seven attempts in all competitions and puts another major dent in Moyes' debut season at United.
Gerrard feels the margin of victory could have been greater, but warned of dwelling on the result.
"I've come here many times and this is one of the most difficult places to come. To come and dominate from start to finish, we're disappointed not to have scored more goals," he added.
"It's a fantastic day personally but we came here to win the game. We'll enjoy it but we need to move on quickly.
"We totally bossed the middle of park and I'm sure if you ask Manchester United defenders, I'm sure they'd tell you they've never had a more difficult day."
