After a pedestrian 45 minutes, both sides showed more adventure in the second half and the Owls took the lead on 83 minutes with a Jacob Murphy goal.
But Tosin Adarabioyo headed in an equaliser five minutes later, before a stoppage-time winner from John Buckley.
Substitute Fernando Forestieri thought he had grabbed a late, late equaliser but his long-range strike smashed against the post.
The Owls arrived at Ewood Park on the back of a four-match unbeaten run.
Monk’s first four away games in charge at S6 had seen the Owls won twice – at Huddersfield Town and Middlesbrough - drawn at Cardiff City and lost at Hull City.
Wednesday’s last visit to Ewood Park saw Jos Luhukay’s team lose 4-2 in December 2018. But that was the only time in their five previous meetings the Owls have failed to get the better of their Lancashire rivals.
The visitors suffered a blow before kick-off, with midfielder Barry Bannan ruled out injured, meaning a recall for Kieran Lee.
German centre-half Julian Borner returned from injury, meaning Sam Hutchinson – captain with Bannan and Tom Lees missing – moved back into midfield, and Atdhe Nuhiu dropping to the bench.
And it was the Owls skipper who had the first chance of the afternoon, heading Joey Pelupessy’s far-post corner over the crossbar.
It was the first of nine Wednesday corners in the opening 45 minutes, but the visitors – without the set-piece expertise of Bannan - failed to create any clear openings.
Kadeem Harris saw his fierce drive blocked at his near post by Rovers goalkeeper Christian Walton, while at the other end, Keiren Westwood palmed over Sam Gallagher’s header.
Wednesday’s best chance of the opening half, saw Dominic Iorfa escape the home defence, from Pelupessy’s free-kick, but his header was hoofed off the line.
Steven Fletcher – who was often isolated up front in the opening 45 minutes – had two half chances to break the deadlock.
First he volleyed wide from 25 yards, then his header – from Morgan Fox’s cross – had Walton scrambling back, but the ball cleared the crossbar.
Rovers should have taken the lead at the start of the second half. Lewis Holtby set Gallagher free, whose original shot was blocked by Borner, before Westwood denied Lewis Travis from close-range.
Gallagher then appeared to be tripped by Hutchinson, but referee Andy Woolmer waved away Rovers’ penalty appeals.
Wednesday brought on Forestieri to provide much-needed support to Fletcher, while Monk also switched the ineffective Adam Reach for Murphy.
Westwood twice came to Wednesday’s rescue. First, he was at full stretch to thwart Adam Armstrong.
Then, from Armstrong’s cross, substitute Joe Rothwell saw his header tipped over the crossbar by the Republic of Ireland stopper.
The game was now end-to-end – in stark contrast to the first half – and Harris cut in from the left flank, but Walton saved his low, curling shot.
Nuhiu – brought on as Monk sniffed a winner – saw his header clear the bar, and the goal finally came in the 83rd minute.
Fox’s cross was headed towards goal by Fletcher, and the rebound dropped for Murphy to tuck away the ball at the back post.
But two late goals turned the match on its head.
A free-kick to the far post saw Adarabioyo rise highest and his looping header beat Westwood.
Then, the Owls goalkeeper was left stranded – after a rare error from Borner - as Buckley’s deflected shot, in stoppage time, ended up in the back of the net.
There was still time for Forestieri to strike the woodwork, but Blackburn managed to hold out for their first win in seven.
Boss Garry Monk said his team paid "the ultimate price" as their winless streak on the road stretched to three.
"We knew coming here would be tough," he said. "We had the lead, the control but for the last five minutes we lost concentration and paid the ultimate price.
"We were exactly where we needed to be for the game we prepared for, but mistakes proved costly.
"We worked so hard but you can't make those mistakes at this level. It's hugely disappointing to give up those three points as we did.
"To not sit out those final few minutes, we'll learn from it and take it on the chin, making sure it doesn't happen again.
"They are experienced players in there, hugely disappointed.
"We prepared and worked hard in the week, we got the game we thought we would get here but switched off in the last five minutes. We have to take responsibility.
"It's about tidying up those little things now and dusting ourselves down. The players are giving their all."