The 21-year-old struck twice in the first half, reviving memories of Wayne Rooney's United debut 11 years ago when he netted a hat-trick in a Champions League game against Fenerbahce.
Memphis also provided a superb cross to set up the stoppage-time goal by Marouane Fellaini that sealed United's win in the first leg of their Champions League playoff.
"When you score a goal then you are the hero," United manager Louis van Gaal told reporters.
"I want to kiss him tonight, because when you score two goals and give an assist also, then normally in the world you are the hero."
Memphis, who was deployed in his favoured starting position on the left rather than as a traditional No. 10, showed the kind of form that led United to pay PSV Eindhoven a reported 25 million pounds for him in May.
Memphis struggled in his first two Premier League games but his contribution against Bruges hinted that he can handle the pressure of the iconic number seven jersey, previously worn by United greats like George Best, Eric Cantona, David Beckham and Cristiano Ronaldo.
"I'm happy that I scored and of course I'm happy that we won," said Memphis, who missed two good opportunities to complete his hat-trick.
"But I must admit I'm a bit disappointed because I could have scored the last two chances and got a hat-trick.
"I will think about that tonight and forget about it tomorrow."
Van Gaal acknowledged his compatriot's hunger to succeed.
"He's never satisfied," he said. "I like that. You have to know what you are doing and also what you can do and then your desire to improve is higher. And that's Memphis."
United full back Luke Shaw, who linked up well with Memphis down the left, also praised his team mate's performance.
"Now he's off the mark, hopefully it can bring him many more goals," Shaw said.
"I think our relationship is starting to build up. We have played a couple of games together and I am starting to understand how he wants to play," he added
"It is great to be playing with him, he can beat a man and he likes the attacking support from me as well," the 20-year-old said.
(Additional reporting by Simon Jennings in Bangalore; editing by Sudipto Ganguly)
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