The Red Devils were kept at bay by a disciplined Cottagers outfit before the in-form striker broke the deadlock to hand his side a 10-point lead at the top of the Premier League.
And, after the game, Sir Alex claimed that the result had been merited even after the visitors were forced to overcome a late onslaught.
"We had so many chances,” he told ESPN. "In the first half we hit the woodwork three times but we deserved to win.
|5/2||Manchester United are 5/2 to win the FA Cup with BetVictor|
The 71-year-old went on to express delight at his side's sizeable lead over Manchester City, insisting it was down to the determination amongst the squad.
"It's a good position [we are in]," the Scot continued.
"We are pleased. Tonight you can look at the performance of the players. They put their bodies on the line, blocking shots, great headers out of defence, really determined effort by the players."
Meanwhile, winning goalscorer Rooney also admitted that Martin Jol's men had made life "difficult" for United.
"We knew it was going always going to be difficult coming here and they made it difficult for us," he said. "We hit the post and crossbar three times.
"We had to keep going and believe we could get the goal, and thankfully we did. We know we can create chances and we can take them. We are in a great position but we are not going to get carried away."