Real coach Zinedine Zidane left out Cristiano Ronaldo and Gareth Bale in a largely second-string side but not even the injection of youth could breathe life into his team until Asensio's late goal ended a run of three games without a win.
Real struggled against a weakened Leganes team showing eight changes from the 3-2 La Liga defeat by Real Betis on Monday, indicating the host' lack of enthusiasm for the Cup despite supporters packing out their Butarque stadium.
Mateo Kovacic blasted Real's best chance wide before halftime and goalkeeper Kiko Casilla swatted away a Claudio Beauvue volley to deny Leganes shortly before Spain international Asensio met Theo Hernandez's cross at the near post and buried it low in the bottom corner.
"We can't say that we played very well but we got a result which is very important to our chances of winning the tie," Zidane told reporters.
"We now need to go on a winning run, we have to be patient and believe that we can do it if we all pull together. This win is a chance for us to string two, three or four wins together."
The second leg at the Santiago Bernabeu is on Jan. 24.
(Reporting by Richard Martin, editing by Ed Osmond)
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