The England midfielder struck United's second goal after 11 minutes and followed that up with a second-half third as Arsenal were pushing for an equaliser.
Mourinho was delighted with the goals but all pointed to his all-round performance for the team, and says he has taken on board what they worked on in pre-season.
"He played well," United boss Mourinho said. "He did what the team needed him to do, defensively and also in terms of going into finishing areas.
"We worked pre-season well and hard about playing sometimes with three at the back, sometimes four. When we play with two strikers we need people from the midfield that can arrive in finishing situations and that's what Jesse did for us."
Greater responsibility may fall on Lingard in the coming weeks when Paul Pogba will be suspended for second-placed United following his red card against the Gunners.