Zlatan Ibrahimovic got the second of the afternoon for Manchester United
Blackburn started well and Marvin Emnes tested Sergio Romero, firing a shot from distance which the Argentine pushed away.
And it was the Championship side who got the first goal of the game when Emnes released Danny Graham who struck past Romero into the far corner.
United almost got level when Ander Herrera struck at goal, but his shot was straight at Jason Steele.
Danny Graham got the opening goal for Blackburn
But the Red Devils were level when Henrikh Mkhitaryan sent Rashford through, and the striker knocked the ball past Steele before rolling it into an empty net.
Jesse Lingard almost gave Jose Mourinho's side the lead just before half-time, but his glancing header went just wide of the post.
After the break substitute Ibrahimovic almost made an immediate impact, but he couldn't quite reach Herrera's low cross.
Marcus Rashford drew Manchester United level
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Soon after the Swedish striker did find the back of the net to put United ahead after some good work from Paul Pogba.
Blackburn thought they had a late equliser when Anthony Stokes had the ball in the back of the net, but the goal was ruled out for offside.
And United held on to book a place in the FA Cup quarter-finals.