Rostov were on the back foot from the outset but were beginning to give United the jitters before Mata ended a move he started, securing a 1-0 second-leg win that sees Mourinho's men through the last-16 tie 2-1 on aggregate.
But as the visitors became more adventurous, United hit them on the break, Ibrahimovic flicking the ball into Mata's path for the midfielder to fire home from close range.
Aleksandr Bukharov, Rostov's scorer in the first leg, threatened to peg United back in the 79th minute, but Romero was equal to both his glancing header and Noboa's unsafe stoppage-time free-kick.
PAUL POGBA sunk to the deck holding his hamstring during Manchester United's Europa League clash against Rostov.
The pitch was a nightmare at Rostov but I think it could be one of those slow nights where United are sat on a narrow lead before finishing them off. "We have the same players in the last match plus Ibrahimovic returning for the Europa League".
Pogba has scored seven goals and provided five assists in 41 appearances over all competitions for United this season.
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The baby's mother, Bridget Alex, said, "When I came and picked up my son she [babysitter] was still on hold with 911". Brandon's babysitter said the boy had fallen off a daybed and was barely breathing while she tried to reach 911.
Rostov weren't without their threat, either, and Sardar Azmoun tested the skills of Romero before coming off to be replaced by Aleksandr Bukharov.
But just a few minutes later and Mourinho's men were attacking down the other end, with Antonio Valencia swinging a cross into the box that Ibrahimovic could only head over the bar.
The Reds had to make another enforced adjustment as Blind came off shortly after receiving treatment, with Phil Jones entering the fray and Marcos Rojo switching to left wing-back. It took a little while in terms of match time, but Ibrahimovic was back to doing Ibrahimovic things in short order.
A Christian Noboa free-kick was brilliantly saved from Romero to stop the game going into extra-time.
Subs not used: De Gea (GK), Young, Carrick, Lingard, Rashford.