Espanyol hadn't won since Nov. 5 and would have set a record for longest losing run in the history of the Barcelona-based club with another defeat.
Iglesias scored with a ninth-minute header after David Lopez's long strike hit the crossbar.
Espanyol forward Leo Baptistao squandered two good chances to put the result beyond doubt, but the hosts' defense kept Leganes, which was unbeaten in seven games, from seriously threatening.
Leganes' Gerard Gumbau was sent off with a second yellow card in stoppage time for a dangerous high kick on Hernan Perez.
"It wasn't our best day today, but we were coming off a bad streak that we probably didn't deserve," said striker Iglesias after scoring his ninth league goal.
Also, defender Jorge "Coke" Andujar scored late to give 10-man Levante a 2-2 home draw with Girona.
Girona led twice and had the numerical advantage from the 69th after Sergio Postigo received his second booking.