United beat Olympiakos 3-0 in the UEFA Champions League on Wednesday to reach the quarter-finals on the back of a 3-0 home defeat in the Premier League by Liverpool last weekend.
David Moyes' side followed up the European triumph with an impressive display at Upton Park as Rooney's first-half brace clinched a ninth away win of the campaign.
Rooney, whose opening goal was scored from just inside the West Ham half, was delighted with the result which was achieved with midfielder Michael Carrick playing at centre-half.
"Today was a good result and I think a much-needed result after last week," Rooney, who captained the team in the absence of the suspended Nemanja Vidic, and with Patrice Evra on the bench, told Sky Sports 1.
"The win on Wednesday night against Olympiakos gave us all a big lift and you can see today the confidence is running through the team. It was a good victory and I'm really pleased with that.
"I think the Olympiakos result was massive for us in terms of picking the fans up, picking ourselves up after the defeat to Liverpool.
"Today we played some lovely football. We're a bit disappointed not to take more chances to make the game a bit easier.
"Overall, we played some good stuff, we defended excellently and I thought Michael Carrick at the back, in such a difficult game, was outstanding for us, so there are lots of positives from today."
Regarding his goal, Rooney added: "I think it's just instinct. I've turned, had a quick look and seen the goalkeeper is off his line. I've tried that many times but thankfully it's gone in.
"You try them in training, you try it in games, but it's not often they come off. When they do it's great for me and the team to put us in such a position."
United now face local rivals Manchester City at Old Trafford on Tuesday night and Rooney is looking forward to the game on the back of consecutive wins.
He continued: "Of course, it's a massive game. It's a big game for the city, for the fans, for the players, and I'm looking forward to it.
"It's a big challenge for us, but we have to go into the game confident after the last two games. I'm looking forward to it, excited, and I hope we can get a good result."