However, dampening the mood was the injury sustained by captain Paul Pogba, who was taken off after just 18 minutes following an apparent hamstring niggle.
Fellaini calmed the nerves around the stadium when he opened the scoring after 35 minutes, nodding home Ashley Young's inch-perfect cross from close range.
Lukaku doubled the advantage shortly after the restart, rising to meet Daley Blind's cross and heading home with power.
United were untroubled in the second half, but a mistake from Victor Lindelof nearly allowed Mohammed Elyounoussi to pull one back from the Swiss side, however, David de Gea's strong hand denied his short-range effort.
19-year-old Rashford made sure the result was sealed six minutes from time, finishing off Fellaini's deflected cut-back. His goal made him the fourth United teenager to score in the Champions League.