Diego Simeone's side succumbed 1-0 to Real Mallorca last weekend and was eliminated in Europe by Manchester City on Thursday (AEST), before struggling again at Wanda Metropolitano.
Atletico had to wait until the 52nd minute for substitute Carrasco to strike with the hosts' first shot on target, before Raul de Tomas levelled with a free-kick after Geoffrey Kondogbia's dismissal.
However, a late handball decision against De Tomas offered Atletico a penalty, which Carrasco converted in the 10th added minute to send Simeone's side level on points, at least briefly, with Sevilla and Barcelona, and three clear of fifth-placed Real Betis.
Atletico had defeated Espanyol in seven of their past nine home league encounters, but it was Vicente Moreno's visitors that started the brighter.
Leandro Cabrera's header from Darder's corner forced a sharp Jan Oblak stop just seven minutes in, while Joao Felix skewed over on the counter-attack at the other end.
Simeone responded by sending on Antoine Griezmann, Matheus Cunha and Carrasco at half-time and Atletico's fortunes soon changed.
Oblak reacted quickly to deny Darder and that save proved vital a minute later when Cunha and Carrasco combined on a clinical break.
Cunha raced towards the Espanyol goal before laying an inch-perfect pass for Carrasco to cut in and squeeze a right-footed strike under Diego Lopez to open the scoring.
Oblak then thwarted a low De Tomas volley but, after Kondogbia was shown a second yellow for handball, the Espanyol striker's resulting free-kick squirmed under the Atletico goalkeeper.
Carrasco snatched victory after De Tomas was adjudged to have handled an Atleti corner, allowing the home forward to fire into the bottom-left corner from 12 yards.