Jose Mourinho's side will now hope to go on and win their second trophy of the season after securing the EFL Cup with a dramatic 3-2 win over Southampton last month.
Rostov were struggling to string passes together with the hosts very much in control midway through the first half. An outswinging corner by Daley Blind was met by a powerful Marcos Rojo header, though Rostov keeper Nikita Medvedev was on hand to deny both the Argentine and Zlatan Ibrahimović on the follow-up. With the angle tight, the Swedish striker was unable to find the net and his shot scrambled wide.
Ibrahimovic was then kept out by the woodwork again 11 minutes before the break, drilling a fierce effort onto the right-hand post after cleverly sidestepping Miha Mevlja in the area.
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United were forced into an early change just three minutes after the restart as Pogba came off with an injury and was replaced by Marouane Fellaini.
Rostov barely threatened in the opening exchanges but they almost took the lead in the second half when Sardar Azmoun forced a save from United stopper Sergio Romero.
United heeded the warning and pressed hard for the opener, with Ibrahimovic heading over from Antonio Valencia's cross.
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The resurgent visitors, now seventh in the Russian Premier League, began to offer more of a threat which opened the game up in United's favour and the Reds' patience was rewarded in the 70th minute.
But United took a lead that their first-half play deserved as Mata slotted home from close range, on 70 minutes, after Ibrahimovic's back-heel flick from a good, low Mkhitaryan cross from the right fell kindly for the Spaniard at the back post.
That chance marked the switch to a more purposeful set-up from the away side, but that only served to give United more space to attack.
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There was more drama to come with Celtic denied a clear penalty just seconds later as Leigh Griffiths was brought down in the box. A late Clint Hill equaliser helped the Gers become only the second team to take points off the champions this season.
Rostov came inches from levelling on the night and on aggregate just eight minutes later.