Rovers, whose priority is to avoid relegation to English football's third tier, took the lead in the 17th minute through Danny Graham but Marcus Rashford levelled before half-time.
Twelve-times FA Cup winners United dominated the game after the break at Ewood Park but had to wait until the 75th minute for Ibrahimovic to slot the winner after being set free by a precise long pass delivered by fellow substitute Paul Pogba.
It was Swede Ibrahimovic's first goal in the FA Cup and 24th in all competitions this season.
