The United defender took a moment to replenish himself during the second half of a 1-0 victory which sent his side through to the competition’s quarter-finals.
Rojo was handed a peeled banana by substitute Ashley Young, and Mourinho later explained that his defender needed to eat to combat fatigue.
“I was tired many times,” he told BT Sport after the game. “I know about the body needing anything, and the player asked for a banana.
“The banana situation isn't funny at all, we have to respect a player's physical situation.”
United’s narrow win over Rostov was their third game in eight days, following the draw away from home in the first leg and Monday’s FA Cup quarter-final defeat to Chelsea.
Mourinho lamented the fixture schedulers after the game and suggested that his side’s Premier League rivals, namely Chelsea and Liverpool, whose calendars are significantly clearer.
“It's bad for us as all of the teams above us don't have [European] football. They do not know what fatigue is. We fight until the limits and that's what we have to do.”
Rojo is not the first United defender to enjoy a mid-game snack. Rio Ferdinand once enjoyed a Jaffa Cake during a Premier League meeting with Charlton Athletic in May 2006.