Unai Emery's Valencia side - who came off a 4-0 loss to Espanyol in La Liga - were lucky to escape from the Spanish capital having somehow scored twice from set-pieces despite doing little else of note on a night when the home team could have scored more than four goals.
Colombian Falcao turned a lovely header into the net from 10 yards out on 18 minutes after Arda Turan had dinked the ball across goal with Adil Rami failing to shepherd it out of play.
Falcao has now scored 27 goals in 27 matches in the Europa League while Atletico are the first team to win 10 straight matches in the competition, but manager Diego Simeone will rue losing two goals on the night.
Jonas scored the equaliser a couple of minutes into stoppage time in the first half against the run of play when he pounced to nudge the ball over the line from Rami's header from a Tino Costa corner.
Atletico moved 2-1 clear on 49 minutes when Miranda popped up to head home from Diego's free-kick before Adrian benefited from some slack play by Mehmet Topal to beat goalkeeper Diego Alves with a smart finish after an impressive run on 54 minutes.
Falcao seemed to have completed a convincing 4-1 win when his shot cracked off the underside of the bar before nestling in the net on 78 minutes from the edge of the visiting area only for Ricardo Costa to give Valencia fresh hope when he netted with a header from Tino Costa's delivery in the final passage of play.
Atletico had several other opportunities to score with Diego Alves blocking a shot by Diego on six minutes with the Valencia keeper halting an effort from Adrian moments later.
Rami showed glimpses of his threat when he powered a header wide of goal on 32 minutes, but Alves was needed to stop a shot from Diego hitting the net on 66 minutes before halting the rebound by Arda Turan.
He then threw himself across goal to swat Diego's free-kick over his bar midway through the second period with Atletico - winners of this tournament in 2010 - camped in Valencia's half.
The return leg will be staged at the Mestalla next Thursday with the home side needing a 2-0 win to reach the final.