Before Ibrahimovic's intervention, 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 halftime.
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.
Blackburn thought they had scrambled a late equaliser when Anthony Stokes poked the ball home but a linesman's flag cut short the celebrations.
Earlier, Harry Kane's hat-trick turned Tottenham Hotspur's potentially tricky tie at Championship side Fulham into a Sunday stroll by the River Thames as they sauntered to a 3-0 victory at Craven Cottage.
England striker Kane struck early on and twice after the interval to take his tally for the season to 19 as the eight-times FA Cup winners eased into the quarter-finals.
It was Kane's second hat-trick in his last seven games and he has now bagged 20 goals in his last 22 London derbies.
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Tottenham Hotspur's Harry Kane completes his hat-trick against Fulham at Craven Cottage REUTERS
Stung by consecutive defeats against Liverpool in the Premier League and Gent in the Europa League and wary of becoming another weekend Cup shock, Tottenham manager Mauricio Pochettino fielded a strong line-up and was rewarded.
With central defender Jan Vertonghen back from an ankle injury after missing five games and Kane having shaken off a knock sustained in the 1-0 defeat in Belgium on Thursday, Tottenham pinned Fulham back into their own area.
Spurs took the lead in the 16th minute with a move of simplicity. Right back Kieran Trippier, in for the rested Kyle Walker, released Christian Eriksen with a long throw and the Dane crossed low for Kane to volley home at full stretch.
He doubled Tottenham's lead in the 51st minute, again glancing in from a pinpoint Eriksen assist, and finished off Fulham in the 73rd minute when he ran unopposed on to Dele Alli's clever pass and tucked a right-foot shot low past Fulham goalkeeper Marcus Bettinelli.
Fulham, still with an outside chance of making the Championship playoffs, had their moments.
Floyd Ayite come agonisingly close to getting a touch to Scott Malone's low cross just before halftime while substitute Scott Parker, a former Tottenham player, headed straight at Michel Vorm late on as Fulham sought some consolation.
Pochettino is yet to win any silverware with Tottenham and clearly is eyeing the FA Cup as well as the Europa League.
Fulham 0 Tottenham 3
(Kane 16, 51, 73)
(Graham 17) Man Utd 2
(Rashford 27, Ibrahimovic 75)
Sutton v Arsenal, 1:55am
Chelsea v Man Utd
Middlesbrough v Huddersfield/Man City
Tottenham v Millwall
Sutton/Arsenal v Lincoln