The 33-year-old dispatched penalties either side of half-time to give the visitors a comfortable lead, but he clipped the post with a third as Nemanja Vidic was dismissed before Luis Suarez wrapped things up six minutes from time with a smart finish.
The win saw Brendan Rodgers' side move to second in the Premier League, four points behind table-topping Chelsea, but with a game in hand and fixtures against Manchester City and the leaders still to be played at Anfield.
Liverpool are unbeaten in the league throughout 2014, winning eight of their 10 outings, and have real hope of securing a first league title in 24 years.
At this stage last season, United were 29 points ahead of Liverpool. 14 now separate the two sides and Gerrard believes this is a reflection of how the tides have turned.
"I've come here many times and been played off the park. They're a fantastic team," he told Sky Sports.
"This is one of the most difficult places to come in the Premier League so to come here and dominate from start to finish, we're still going away slightly disappointed that we never got more goals.
"It's a fantastic day personally, getting a couple of goals. We came here to win the game.
"I said before the game it's difficult and we need to just take each game as they come and try and win them all.
"We'll enjoy tonight. We need to move on very quickly because Cardiff's going to be tough.
"I think we've shown today that we're genuine contenders and we're going to fight to the end for this."
Gerrard also hailed Liverpool boss Rodgers, who made one change from the 3-0 win at Southampton, with Raheem Sterling coming in for Philippe Coutinho in midfield.
"I think you've got to give credit to the manager," Gerrard said. "He keeps switching it, tinkering with the formation and the tactics. We played a diamond today and totally bossed it in the middle of the park.
"When you come away from home you've got to keep the ball and dominate the middle of the pitch and we know if we do that, we've got two of the best strikers in the world to do the damage.
"I'm sure if you ask the Manchester United defenders today, I don't think they've ever had a harder 90 minutes than that."
Speaking of his three penalties, Gerrard added: "It was strange. I maybe got a little bit cocky with the third one but I was confident I was going to score and I'm very pleased with the two."
Earlier in the season, Liverpool were keen to play down their title chances, insisting that Champions League football was their priority, but Gerrard says they have their sights firmly set on silverware.
"We believe it but I've said it many times, in football, the easiest thing is to talk the talk and we've got to go and treat Cardiff like Manchester United and then Sunderland like Manchester United," he said.
"And if we do, we'll win a lot of games between now and the end of the season so the rest of the people that are fighting for it with us have got to believe we're going for it."
Boss Rodgers has previously claimed that the Reds were not in the title race and, although he was more positive after the Old Trafford victory, he opted to stay cautious when talking about silverware.
"It's exactly the same," he told Sky Sports. "I think we're a developing side. We're probably a year ahead of where we've been but that's testament to the players and how quickly they're adapting to the methods.
"The biggest difference is that we're believing all the time. You come here, you win 3-0 playing like we did today and it's another statement in terms of where the team is going.
"I repeat (what I said) before, we could win the next nine games and still not win it but certainly our focus now will be to get ready.
"We've got a tough game against Cardiff next week away and it's five games on the spin now. Terrific results against tough opponents and we've just got to keep it going.
"I think that we're coming into a real strong stage of the season. We did the same last year and we're doing the same this year."