Club Atlético de Madrid will take a slender advantage into their UEFA Europa League quarter-final second leg at Hannover 96 next week after Eduardo Salvio's late winner.
The substitute, who registered twice at the Vicente Calderón as Atlético beat Beşiktaş JK in the last 16, curled a shot beyond goalkeeper Ron-Robert Zieler with a minute remaining. Hannover thought they had done enough to earn a draw after Mame Diouf had cancelled out Falcao's early opener.
Atlético had won their previous seven home games in Europe and started this one strongly, too, Falcao forcing a block from Zieler. The two were again involved in the ninth minute when the Colombian international registered the Liga team's 400th goal in UEFA competition, rising to beat the goalkeeper to Gabi's floated free-kick around the penalty spot.
The hosts pressed forward in search of a second, with Adrián López dragging a shot wide following some good work from Arda Turan, whose chipped pass then teed up Falcao. Last season's 17-goal UEFA Europa League top scorer should have done better than head over at the far post.
Competing in their first European quarter-final, Hannover equalised seven minutes before the break. Spotting Lars Stindl's run down the right, Steven Cherundolo lifted a pass into the 23-year-old, whose inviting pass across the area fizzed past the watching Miranda and into the path of Diouf, who slid in his third goal in four appearances in the competition.
Falcao shot wide as the Rojiblancos strived to regain the initiative in the second half. Their bright start was short-lived, however, as Mirko Slomka's side almost grabbed a second when Diouf connected with substitute Didier Konan Ya's right-wing cross; Thibaut Courtois intervened with an instinctive one-handed stop.
With the tie heading for a draw, Salvio pounced on Falcao's header and bent a shot high into the net. It could have been even better for the locals had Diego's spectacular shot not been well saved by Zieler in added time. The night was not totally positive for Diego Simeone's side, however, with Juanfran, Gabi and Arda receiving bookings which rule them out of next week's return.