Rodriguez set them on their way to victory with an early goal before Morata scored twice in the space of five minutes to put Real 3-0 up.
Leganes, however, grabbed two goals in quick succession through Gabriel Appelt Pires and Luciano to reduce the deficit to 3-2 at halftime.
An own goal from Leganes defender Martin Mantovani stretched Real's lead at the start of the second half and they kept their hosts at arm's length for the remainder of game to recover top spot from Barcelona, who had provisionally led the standings after beating Sevilla 3-0 earlier on Wednesday.
Real top the table with 71 points, two clear of Barca, and have a game in hand on their arch rivals as they bid for a first title since 2012.
(Reporting by Richard Martin; Editing by Toby Davis)
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