Watford came into the match with four wins in as many games this season.
But the Hornets found themselves 2-0 at half-time thanks to goals from Lukaku and Chris Smalling.
The home side grew into the match in the second half, hitting back through Andre Gray and they pushed for an equaliser but with almost the last shot of the game, Christian Kabasele’s header was tipped away by De Gea.
And after the match, Mourinho praised Lukaku for his defensive effort and called De Ge the best in the world for his save.
He said: “I think this last Romelu Lukaku tackle as a right-back shows everything.
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“I think he is the image of the team, a team that played very well in long periods in the first half.
“But we had the experience also here last season where they don’t give up and when the game is open they believe.
“That is the reason why De Gea is the best, why he is Manchester United’s goalkeeper and when you are goalkeeper of a big team, even if you don’t touch the ball for the whole game you have to be decisive.”
United’s win puts them level on points with Spurs and Arsenal and Mourinho said his side had to keep up with their rivals.
Mourinho said: “With this win we keep the same points as Spurs and Arsenal.
“We have that distance of four and six points to the other top teams, if we don’t win this match obviously that distance goes a little more.”
Lukaku’s goal was his fourth of the season and Belgian said United were guilty of sloppiness at the start of the match at Vicarage Road.
He said: ”Credit to Watford. We were too sloppy but it’s another win and we have to keep going.
“We had a few good runs from Lingard, Sanchez and myself, I scored, Chris [Smalling] scored, we had to try and score a third but we were sloppy, we have to look again at the images.
“The manager said to us we were playing well but not great, he made us humble at half time.
“We ground the result out, that’s what we need to do, to catch the teams above us.”