Rovers put in a gallant effort but they went out of the Emirates FA Cup after losing 2-1 to Manchester United.
A late strike from Zlatan Ibrahimovic proved to be the difference as the visitors sneaked into the quarter-finals.
Rovers stunned their illustrious opponents by grabbing the opening goal, Danny Graham's sumptuous strike sending the home fans into raptures.
But the scores were level on 27 minutes when Marcus Rashford raced through before tucking home.
It continued to be an even contest in the second half with little between the two sides. Rovers thought they had regained the lead when Charlie Mulgrew poked home, but the whistle blew for an infringement.
United threw on Paul Pogba and Ibrahimovic in an attempt to win the tie and the two would combine to great effect 14 minutes from time. The Frenchman's long ball over the Rovers back line saw the Swede in on goal, and with a deft flick of the boot it was in the corner of the net.
Rovers never gave up and only a triple save from Sergio Romero stopped them from deservedly earning a replay. The United keeper spilled Connor Mahoney's initial shot, but he redeemed himself to thwart Anthony Stokes and Marvin Emnes on the follow-up. The ball did end up in the back of the net, but Stokes was a yard offside.
Having pushed United all the way the home side can be proud of their performance. It's back to the league this Friday night when they take on Burton Albion.
Ahead of the game Rovers made four changes from the side that faced Sheffield Wednesday in midweek. Three of them were enforced as injury and suspension ruled out Sam Gallagher, Hope Akpan and Elliott Bennett, whilst Connor Mahoney dropped to the bench.
Gordon Greer returned from injury, whilst Marvin Emnes, Liam Feeney and Craig Conway all came into the starting eleven.
Jose Mourinho's side saw seven changes from the side that defeated Saint Etienne on Thursday and it was they who saw plenty of the ball in the early stages without ever imposing themselves in the final third.
In contrast it was the Rovers who were the first to have an effort on target on 16 minutes when Marvin Emnes stung the gloves of United keeper Sergio Romero with a sweetly struck shot from the edge of the box.
A minute later the dreadlocked Dutchman would play a key role in Rovers grabbing the opening goal of the game.
The forward lead Marcos Rojo a merry dance as quick feet and tricky saw him bamboozle the defender before slipping the ball into the path of Danny Graham who blasted into the top corner of the net with a superb left-footed strike to send the home fans into raptures.
Such was the quality of the move and finish, opposition boss Jose Mourinho stood and applauded.
That woke the visitors up, Ander Herrera seeing a shot palmed away by Steele before Henrikh Mkhitaryan fired wide from distance.
But on 27 minutes the Premier League side drew level. A free-kick up to the edge of the penalty area from Rovers was cleared and in an instance Mkhitaryan sent a defence splitting pass through for Marcus Rashford. The young striker showed a cool head to round the advancing Steele before stroking the ball into the back of the net.
Despite the disappointment of conceding an equaliser, Rovers continued to carry a threat in the final third. A through-ball from Charlie Mulgrew saw Craig Conway racing into the area but left-footed he put his first-time shot narrowly over.
Then as Derrick Williams nodded down a deep cross Mulgrew let fly but the ball deflected just wide of goal to ensure the two teams would be tied at the break.
Conway saw a low shot saved from a Feeney centre as the second half got underway, but that was a rare opening in what was proving to be a tight contest.
A lack of chances for the visitors prompted a double change on 62 minutes as the formidable figures of Paul Pogba and Zlatan Ibrahimovic were introduced. It almost had an immediate impact as Poga fed Herrera but his ball to the far post was just beyond the outstretched boot of the Swede.
At the other end Feeney flashed a cross that just evaded Emnes, before the Rovers winger should have done better when he met a centre from Mulgrew, but he couldn't get enough purchase on his header and the chance was gone.
Rovers did have the ball in the back of the net on 74 minutes when a corner from substitute Connor Mahoney found Mulgrew, but the Scot was adjudged to have fouled Chris Smalling in his attempt to win the initial header ball.
But with 14 minutes to go it was a moment of magic from the two United substitutes that would eventually settle the tie in their favour.
Beating the offside trap Ibrahimovic collected a long ball upfield from Pogba and with a deft flick he expertly found the far corner of the net.
Anthony Stokes was introduced, as was Willem Tomlinson for his Rovers debut, and the home side were unfortunate not to take the tie to a replay at Old Trafford in ten days time when they saw a second goal disallowed.
Cutting in from the right Mahoney let fly, Romero spilled his shot but then brilliant blocked both Stokes and Emnes on the follow-up. Eventually Stokes forced the ball home but he was in an offside position.
The Rovers pressed until the end but they couldn't get the second goal everyone of a blue and white persuasion craved. United's reward is an away tie at Chelsea in the quarter-finals.
ROVERS: Steele, Lowe (c), Greer, Lenihan, Williams, Feeney (sub Mahoney 73 mins), Guthrie (sub Tomlinson 81 mins), Mulgrew, Conway, Emnes, Graham (sub Stokes 77 mins).
Subs not used: Raya (GK), Hendrie, Nyambe, Brown.
Goals: Graham 17 mins.
Booked: Williams 35 mins.
MAN UTD: Romero, Rojo, Martial (sub Ibrahimovic 62 mins), Lingard (sub Pogba 62 mins), Carrick (c), Young, Rashford (sub Mata 89 mins), Herrera, Mkhitaryan, Darmian.
Subs not used: Henderson (GK), Bailly, Shaw, Schweinsteiger.
Goals: Rashford 27 mins, Ibrahimovic 76 mins.
Booked: Lingard 53 mins, Young 73 mins.
Attendance: 23,130 (including 7283 visiting supporters)