The Colombia striker netted his 21st goal of the campaign in the 38th minute, only to see his captain Gabi sent off for a second bookable offence on the stroke of halftime to leave Atletico with a man less for the entire second period.
Diego Simeone's side stood firm and moved on to 56 points from 25 games in second place, 12 behind leaders Barcelona who came from behind to beat Sevilla 2-1 at home on Saturday.
"We played a good first half, but the second with a man less was almost better," Argentine Simeone told a news conference.
"The effort put in was extraordinary."
Champions Real Madrid are third with 52 points after a 2-1 win at basement side Deportivo La Coruna, also on Saturday.
Fourth-placed Malaga, who lost 1-0 at Porto in the Champions League last week, were well beaten 3-0 at Real Betis to stay on 42 points. Betis climbed to seventh with 39.
Simeone fielded a full-strength side at the Calderon unlike fellow King's Cup semi-finalists Barca and Real, who meet for their second leg on Tuesday and Atletico's opponents Sevilla, who they visit on Wednesday.
Atletico put the visitors under intense pressure, eventually winning a penalty when the outstanding Diego Costa was brought down in the area.
Falcao sent the keeper the wrong way from the spot and with a man less following Gabi's sending-off the hosts outwitted Espanyol in the second half to make it 13 league wins from 13 at the Calderon this season.
Javier Aguirre's visitors, 13th with 31 points, slipped to only their second defeat in 12 outings since the Mexican took the helm from Mauricio Pochettino with the side bottom in late November.
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Betis took the lead against Malaga inside the first minute when Jorge Molina headed in at a corner and they doubled their lead when Mario powered in another header from a corner in the 27th.
With Benat Etxebarria pulling the strings in midfield Betis were rampant, and they should have had their third when Molina's penalty hit the crossbar bounced down and out in the 35th.
Television replays showed the ball had crossed the line.
Dorlan Pabon made it 3-0 from the narrowest of angles before the break, and Malaga's comeback hopes withered when Ignacio Camacho was sent off in the 62nd.
Earlier, Rayo Vallecano's push for European football next season suffered a blow when they lost 2-1 at home to Real Valladolid to drop to eighth with 37 points.
Rayo defender Jordi Amat scored an own goal in the 71st minute but rectified the situation a minute later when he caught the visiting keeper off his line with a spectacular shot from around 40 metres out.
Former Manchester United striker Manucho grabbed Valladolid's winner with 10 minutes remaining.
Elsewhere, new Celta Vigo coach Abel Resino started with a 2-1 home victory over former club Granada
