A 1-1 draw in Russia a week ago had given United the slight advantage heading into tonight's match, and Juan Mata's second-half strike sealed their place in the last eight despite a largely frustrating evening for Jose Mourinho's side.
In contrast, United were able to make five changes to their side, including a recall for Zlatan Ibrahimovic, and the Swede almost broke the deadlock inside five minutes when he poked the ball against the post from an impossible angle after Marcos Rojo had tested Nikita Medvedev with an unmarked header.
It was one-way traffic throughout the opening 45 minutes as United dominated possession, but they struggled to find the breakthrough and Ibrahimovic missed another opportunity to open the scoring when he was denied by a sprawling Medvedev after beating the keeper to the ball on the edge of the box.
Henrikh Mkhitaryan's away goal in the first leg had given United the lead in the tie, but the Armenian failed to hammer home that advantage when he was released through on goal, sending his finish wide when looking to lift the ball over the keeper.
The hosts continued to apply pressure and Ibrahimovic was once again within the width of the woodwork of breaking the deadlock in the 34th minute when he cut inside his man before thumping a fierce effort against the post.
Paul Pogba then fired a free kick over the top before Rostov finally recorded their first shot of the match in first-half stoppage time, although it was never troubling Sergio Romero in the United goal as Alexandr Erokhin sliced a 25-yard effort well off target.
There was one final piece of work for Medvedev before the interval when Pogba curled a shot into the arms of the keeper, who made a comfortable save to ensure that his side went into the break still level.
Despite still being ahead in the tie, the level of United's dominance without reward on the scoreboard was increasingly frustrating for the hosts, and they suffered another blow in the opening minutes of the second half when Pogba limped off with a suspected hamstring injury.
The tenuous nature of their away-goal lead was reiterated 10 minutes into second half when Sardar Azmoun cut inside Rojo before curling an effort towards the top corner that Romero needed to be at full stretch to push away.
It was Rostov's first shot on target of the night, but a second arrived moments later when the resulting corner was only cleared as far as Christian Noboa, who drew a more routine stop from the United keeper with a first-time volley from range.
United responded with a half-chance of their own as Ibrahimovic flicked a header over the bar, but their grip on the game began to loosen as the second half progressed, with Rostov beginning to see more of the ball in better positions.
However, it was during this period that the breakthrough finally arrived for the hosts as a rare defensive error from Rostov was ruthlessly punished. Mata pounced on a loose ball in midfield to launch an attack, and ended up applying the finishing touch from close range after Mkhitaryan's cross had been flicked on by Ibrahimovic.
Rostov's task of finding a goal had not changed despite falling behind on the night, though, and Romero needed to be alert to prevent extra time with just over 10 minutes remaining when he scooped Alexander Bukharov's header around the post.
In the end, it proved to be Romero who kept United in Europe despite their dominance for the vast majority of the match, with the back-up keeper making one final save to deny a stunning 30-yard free kick from Noboa that looked to be heading for the top corner.
That proved to be the final kick of the game as the referee deemed that there was not enough time to take the resulting corner, with Rostov unable to force extra time as United took one step closer to winning the only major European trophy which has so far eluded them.
The Red Devils, who are now unbeaten in their last 22 home games in all competitions, will next turn their attention back to the Premier League when they take on managerless Middlesbrough at the Riverside Stadium on Sunday.