The Cottagers entered the contest on the back of an 11-game winless run on the road, while they last celebrated success on the red side of Manchester back in 2003.
After a tepid first half in which clear-cut chances were at a premium, Fulham came out of the blocks after the interval and deservedly opened the scoring on the 65-minute mark when the ball kindly fell to Calvin Bassey, who lashed home a loose ball inside the penalty area.
Man United thought they had rescued a point in the 89th minute when Bernd Leno parried the ball straight into the path of Harry Maguire, who tapped home from close range.
The Red Devils, roared on by the Old Trafford faithful, pushed for a late winner, but it was the Cottagers who had the final say deep into stoppage time when Alex Iwobi completed a counter-attack, instigated by a powerful run from Adama Traore, with a smart finish into the bottom corner.
Speaking at a post-match press conference, Silva said: "It is clear in my opinion the better team on the pitch won the game. We are the team that created the most chances, that played better and tried to win most of the moments of the game.
"We created a very good number of chances in the first half. They had two or three very good moments as well but the way we started the second half, the first 30 minutes, the way we controlled the game against Man United at Old Trafford - that is not easy to do.
"We scored and after came the normal reaction from Man United, the crowd played a big part in their reaction, and we conceded a bit of an unlucky goal.
"In that moment we showed a strong mentality. It was first a moment for us not to concede the second as Old Trafford was on fire but we showed a strong mentality and even before the 2-1 we had a strong chance from Rodrigo [Muniz].
"It was a really well-deserved three points and the best team won."
Asked about the impact of both Bassey and Iwobi, who were handed their first starts for Fulham since returning from Africa Cup of Nations duty with losing finalists Nigeria, Silva said: "They know really well how they are important for us. This afternoon just showed how much we have missed them.
"They are two really important players for us. I remember the last game they played together, before the Man United game, was at home against Arsenal and we won 2-1 as well, they were really important for us that afternoon.
"We are a team, a group of players, even without Joao Palhinha and Willian, who have both been so important for us, we are much more than [one] player, we have shown that we are a team."
Fulham remain 12th in the Premier League table despite beating Man United, but they have moved to within three points of the top 10, while they are also 12 points clear of the bottom three.
Silva and co will hope to build on their victory at Old Trafford when they welcome Brighton & Hove Albion to Craven Cottage for their next Premier League game on March 2.