England international Rashford came on just after the hour mark and netted three goals in 16 minutes, first running clear to calmly slot in, then smashing in a terrific second, before drilling home a third in stoppage time.
Teenage forward Mason Greenwood scored his first goal in the competition, latching onto Paul Pogba’s defence-splitting pass before drilling a precise, low finish into the far corner.
Greenwood produced another thumping effort straight at Peter Gulasci, while the visiting goalkeeper also kept out Anthony Martial’s poked effort.
But when Martial was tripped in the box with three minutes to go, he converted the penalty to score his first goal of the season.
Ole Gunnar Solskjaer’s side backed up their excellent win at Paris St-Germain in their opening game by collecting another impressive three points to extend their unbeaten run to four games.
United’s backline had a comfortable night, David de Gea keeping out efforts from Christopher Nkunku and Ibrahima Konate.
In the Group H’s other game, last year’s runners-up Paris St-Germain picked up their first points with a 2-0 win at Istanbul Basaksehir.