Manchester United boss Louis van Gaal urged his players to be more clinical in the final third at the end of the first 45 minutes at Boleyn Ground.
The Red Devils battled past West Ham United to set up a semi-final against Everton at Wembley later on this month.
And the Dutchman branded the victory as "fantastic" and a "big result" as United continue their quest to lift a trophy this season.
"It was fantastic – I am happy, I'm happy for the players, the club and the fans," van Gaal explained.
"Of course it's a big result and I think we played very well. We played defensively very well – we didn't give so much away. They lost the ball a lot of times and I believe with a bit more composure we could have finished it much earlier.
"I said at half-time we had to show composure in the third and fourth phase. We did that better but I still believe we could have finished the game earlier. At the end it was, of course, not so nice but we are the winners, so it's fantastic.
"When you see the match, we controlled the game for 85 minutes and then suddenly they scored the goal through the kick of Payet, the head of Carroll and of Tomkins – it was also quality. At that time it changed the match but David De Gea saved us. It was a fantastic night.
"It [Upton Park] is a difficult place to come – they haven't lost here since August; it was difficult because they have beaten all the top teams here.
"After the [Europa League] defeat at Liverpool, the next game was against West Ham in the FA Cup; we were 1-0 down and fought until the end for 1-1. The spirit in the team is very good, you cannot deny that. It was a fantastic result and I'm happy I shall go to Wembley as a manager."