Zinedine Zidane's men, 19 points behind Barcelona in LaLiga, had won one of their last five games in all competitions, amid reports the Frenchman could leave the club at the end of the season, and they looked set to be held at Butarque.
But after Leganes had shone on their first appearance at this stage of the competition, Asensio came up with a cute near-post finish from Theo Hernandez's cross to put Madrid in firm control of the tie.
Madrid were held at home by lower-tier clubs Numancia and Fuenlabrada in previous rounds after first-leg away wins and they struggled to respond to Saturday's 1-0 defeat at the hands of Villarreal, with Mateo Kovacic missing a golden first-half chance.
Jose Naranjo and Claudio Beauvue forced smart stops from Kiko Casilla and Marcos Llorente headed against his own crossbar, adding to the sense of injustice at Leganes' defeat.
Madrid were poor for large parts of the contest, but Asensio's dramatic impact may serve to ease the growing pressure on Zidane ahead of Wednesday's return leg at the Santiago Bernabeu.
GOLAZOOOOOOOOOO! What a goal!!! @TheoHernandez produced an excellent cross from the left and @marcoasensio10 expertly volleyed home, at the near-post, from iust outside the six-yard box!@CDLeganes 0-1 #RealMadrid#RMCopa | #HalaMadrid— Real Madrid C.F. (@realmadriden) January 18, 2018
Madrid were forced into an early change, introducing birthday boy Nacho at the back for the injured Jesus Vallejo, but Zidane's men made a flat start to the game and had to wait until the 23rd minute for a sight of goal, which Raphael Varane headed over the crossbar.
Javier Eraso rifled a long-range drive off-target as Leganes, on a four-game winning streak at home, found a foothold in midfield midway through the first half.
But Madrid should have taken the lead when Kovacic picked Ruben Perez's pocket 25 yards from goal, only for the Croatian to slide his shot tamely wide of the near post.
Gerard Gumbau got a vital touch to deflect Dani Carvajal's drive wide before Lucas Vazquez just missed Asensio's cross on the stroke of half-time.
Casilla had to react sharply to block a close-range Naranjo effort after 56 minutes, Diego Rico nutmegging Carvajal with a superb backheel to send the winger clear.
And Leganes came even closer to breaking the deadlock when Llorente could only direct a free-kick against the woodwork, with Casilla stranded.
Zidane brought on Luka Modric and Isco for more midfield control but Leganes substitute Nordin Amrabat had a more immediate impact, hooking a volley wide moments after his introduction.
With the game becoming stretched, Modric played Vazquez through but Nereo Champagne made a fine block, denying the Spain winger his fourth goal in the competition.
Casilla sharply palmed away a Beauvue volley destined to fly inside his left-hand post with 15 minutes to go and when Kovacic again finished meekly, Leganes looked to have held on.
It was not to be.
Clumsy for the most part, Theo had endured a poor game at left-back but his whipped delivery was perfect for Asensio to get in front of his man to flick home a vital goal for Zidane's Madrid.