Despite Jose Mourinho’s side creating a host of chances, Mata’s second-half strike was the only goal of the game which proved to be just enough to take them into the last eight of the competition although Pogba was forced out of play with an injury after the break.
United went close to opening the scoring after just five minutes when Marcos Rojo’s header from a corner forced a saved out of Nikita Medvedev before Zlatan Ibrahimovic fired the rebound against the post.
Ibrahimovic was involved again 11 minutes later when he latched onto the end of a ball over the top from Mata, but after attempting to round Medvedev, his finish from a tight angle was deflected over the bar by the hand of the Rostov keeper.
United kept up the pressure as they fashioned another opportunity on the half-hour mark when Mata again cut open the Rostov defence with a pass, but this time it was Henrikh Mkhitaryan who was unable to take advantage, sending his chipped effort just wide of the target.
The hosts’ dominance continued as the half wore on but still they were unable to find a breakthrough, with Ibrahimovic being denied by the woodwork again when he smashed a fine hit back off the post after cutting into the box on 35 minutes.
Pogba did his bit to try and break the deadlock on the stroke of half-time when picking up the ball on the edge of the area from Ibrahimovic and firing at goal, but Medvedev did well to turn the strike wide and frustrate United once again.
United began the second half slowly, with Rostov creating their first real opportunity of the game in the 54th minute when Sardar Azmoun bent an effort towards goal from the edge of the area that Sergio Romero did well to turn around the post.
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.
And on 70 minutes, United finally managed to find a way through when Ibrahimovic flicked a Mkhitaryan cross into the path of Mata at the far post, who fired into an empty net from just a few yards out.
Rostov could have found an equaliser in the 79th minute if not for Romero, who was on hand once more to save United, this time producing an excellent stop to divert wide Aleksandr Bukharov’s close-range header, while he also tipped over Christian Noboa’s free kick in the dying seconds.