With their mission to revive the tie firmly set before kick-off, Braga flew out of the blocks back on home soil, cancelling out Ferencvaros’ two-goal first-leg advantage inside the first 15 minutes.
Rodrigo Zalazar was the catalyst for Braga's superb start, as his driving run and low cross was turned home from close range by Horta in the 11th minute – for his 18th goal of the campaign.
The Uruguayan then inadvertently teed up Florian Grillitsch for Braga’s second goal, as his blocked shot fell kindly for the midfielder to steer home from 25 yards on the quarter-hour mark.
Ferencvaros were visibly shellshocked and continued to freeze in the face of constant Braga attacking pressure.
Such was the lopsided balance of play that any third goal was only ever going to one team, and the hosts finally edged ahead in the tie when Gabriel Martinez smartly finished off Pau Victor's through ball to make it 3-2 on aggregate on 34 minutes.
With the contest in their hands after the restart, there was simply no letting up from Braga, as Horta calmly swerved home a fine fourth from the edge of Ferencvaros’ box on 53 minutes.
A frustrated Keane opted for a flurry of substitutions either side of the hour, but it was too late to haul his side back into the game, with Carlos Vicens’ rampant hosts storming over the finish line.
A place in the UEL’s last eight pits Braga against either Panathinaikos or Real Betis next month as Keane returns to the day job of fighting to retain his Hungarian title amid major speculation he could be heading back to the Premier League this summer.
Flashscore Man of the Match: Ricardo Horta (Braga)