It took 34 minutes for United to break the deadlock at the Hawthorns, with Phil Jones eventually firing them in front against the Barclays Premier League strugglers.
Second-half strikes from Rooney and Danny Welbeck then put the match to bed, and the United forward admits it was a result they needed after a run of three matches without a win in the league plus a European defeat to Olympiacos.
"We were hurt after losing to Olympiacos and we wanted to come and show everyone we're a good team," Rooney told BT Sport.
"We'll fight for our shirt and for our place in the team. We played with great energy and we always looked a threat. The goals were well deserved and there maybe could have been more.
"We were not able to get forward sometimes in the first half and we had to fight through that.
"But we got the goal thankfully from Phil Jones and went in at half-time and knew we had to finish the game off. So to get the second was great and I think we saw the game off then quite well."
Team-mate Michael Carrick added: "It was all about the result. We wanted to bounce back, it seems a long time since the last game and we were ready for today.
"It was important, we needed the three points, but the game was exiting. We looked good going forward, we defended well so we can be happy with today."
West Brom, meanwhile, have not won in the league since Pepe Mel took charge in early January and the manager knows any decision on his future is out of his hands as his team battle to avoid relegation
"I don't know that (if I will be replaced), it doesn't depend on me," he said.
"I'm very pleased with the performance of the players and very proud of the performance. They played in adverse conditions from the start.
"The next four games are very, very important for the club, starting with Swansea, and now we have to recuperate the team."