Goals from Danny Graham and Marcus Rashford meant that the game remained level with 15 minutes remaining before Ibrahimovic marked his debut in the competition with his 24th goal of the season after coming off the bench.
Jose Mourinho made seven changes to his side for the short trip to Ewood Park and during the opening stages, it appeared that it could be a decision which would backfire.
Moments after testing Sergio Romero from 25 yards, Marvin Emnes turned provider for Danny Graham who was able to find the far corner of the net from 14 yards after being played in by the on-loan attacker.
Henrikh Mkhitaryan produced a delightful pass through the middle of the Rovers defence, and Rashford kept his composure to round Steele and tap into an empty net.
To Blackburn's credit, they continued to threaten at the other end of the pitch and they could have regained the lead when Craig Conway was played through on the left-hand side of the penalty area, but his shot was always rising and it cleared the crossbar.
Both teams struggled to create anything significant during the opening stages of the second half, but Blackburn managed to create a couple of corners and they started to get on top as the game approached the hour mark.
However, their increasing confidence led to Mourinho introducing both Paul Pogba and Ibrahimovic and they almost made an immediate impact when Pogba teed up Herrera to cross for Ibrahimovic, only to see the Swede fail to get the telling touch at the back post.
With 15 minutes left, the pair joined forces to put United into the lead as Pogba's long ball found the run of Ibrahimovic and he calmly slotted the ball into the corner on the half-volley.
With three minutes remaining, Romero had to make a fine double save as Connor Mahoney and Emnes threatened an equaliser but aside from that scare, United comfortably held on to reach the last eight and continue the defence of the trophy which they won last May.