Steven Gerrard has insisted that Liverpool can win the Premier League while, for the reigning champions, Wayne Rooney described Manchester United's 3-0 defeat to Brendan Rodgers's rampant side as a "nightmare" and among his worst days in football.

Liverpool last won the championship 24 years ago but after scoring penalties in the 34th and 46th minutes – and missing a later one – to make him the club's highest scorer against United, Gerrard said: "We believe that we can win the league. I've come here many times and been played off the park. They are a fantastic team and this is one of the most difficult places to come to, so to come and dominate from start to finish and we are still going away disappointed that we didn't score more goals.

"I said before the game it's going to be difficult [to win the title] and we need to take each game as it comes and just try to win them all. We will enjoy tonight but we need to move on very quickly because we know Cardiff [next] are going to be tough. I think we have shown out there that we are genuine contenders and we are going to fight to the end."

With Luis Suárez completing the rout, Gerrard hailed Rodgers's tactical approach featuring Raheem Sterling playing in an unfamiliar role behind the strikers. "You have got to give credit to the manager," said the midfielder. "He keeps switching and tinkering with the formation and the tactics.

"We played a diamond [formation]. We totally bossed the middle of the park. When you come away from home you have got to keep the ball and dominate the middle of the pitch and if we do that we have got two of the best strikers in the world to go and do the damage. I'm sure if you ask the United defenders, I don't think they have ever had a harder 90 minutes than that."

Of his spot-kicks, Gerrard said: "It was strange I maybe got a little bit cocky with the third one. I was confident I was going to score it but I was very pleased I got the two important ones." Amazingly, the first time Gerrard went to see Liverpool as a supporter, he told Match of the Day, Jan Molby scored a penalty hat-trick in a 3-1 victory over Coventry in the League Cup. Insult was added to injury for United when Nemanja Vidic received a second yellow card for a foul on Daniel Sturridge that gave away the final penalty.

United have allowed a 43-point swing to Liverpool compared with last season and this defeat was a fifth at home in the league. The last time they lost more at Old Trafford was in the 1977-78 season. If David Moyes's team drop one more point they are guaranteed to finish with their poorest return of the Premier League era, having twice managed 75 points, in the 1996-97 and 2003-04 campaigns.

Rooney said: "It's like a nightmare. It's one of the worst days I've ever had in football. It's hard to take. You have to give Liverpool credit – they played well – but it's difficult to take. Nobody wants to lose, especially in this way, in your own stadium. It's not nice. We had a game plan for the second half, which went out the window when they got the second penalty so early on. It made it an uphill battle to come back."

Rodgers heaped more pressure on Moyes, who had suggested Liverpool were favourites before the match, saying: "I was probably surprised before the game when I heard we were supposedly coming to Old Trafford as favourites. I would never say that at Liverpool – even if I was bottom of the league. Anfield is Anfield. We expect to win and we have a mentality that has been developing over 18 months which we expect to win home and away and the belief is in the players and you see that in their game."

When this was put to Moyes, he said: "I thought Liverpool were above us in the league and playing well and any average person would have turned around and said that." Pressed if he was surprised a rival manager would question him, Moyes added: "You are entitled to an opinion."

Rodgers also claimed the job he has done since becoming manager two seasons ago could not be compared with that of Moyes on taking charge of United last summer. "I have empathy in terms of the pressures at both of the two biggest clubs in British football but it is incomparable where we were at. He came in and they were champions with world-class players. We were eighth when I came in and there was a hell of a lot of work to do." said the Northern Irishman.

Moyes, as he has been previously this season, was at a loss to offer reasons for the defeat. "It is difficult to explain it," he said. "I just think Liverpool played well. We didn't play as well as we can and we will work to make that better."