The 23-year-old midfielder arrived from Ajax for a fee rising to £40million at the start of the month and made an immediate impression by coming off the bench to score on his first appearance.
The performance highlighted the need for improvements, with Ole Gunnar Solskjaer calling for introspection after his players fell "short" of expectations on Saturday evening.
"Before the game, I was telling the team: 'Believe in yourselves – don't be scared to play your game.'
"Obviously today we showed it and I'm not surprised by the two games we managed to win."
Zaha, who continues to be linked with a summer exit, added: "Obviously we know what they can do but we wanted to impose ourselves on the game.
"We've got so much pace and attacking threat in the team. We just wanted to utilise it from the beginning."