Garcia headed his side in front after 25 minutes and almost blocked Fernando Llorente's equaliser on the line before scoring a technically brilliant, opportunistic winner 16 minutes from time.
Victory puts defeat in the opening game behind Espanyol, while Bilbao have just one point now from six.
Fifteen minutes had passed before either side had managed a shot but it was almost a successful one as Bilbao's Igor Gabilondo struck from 20 yards.
The ball took a heavy deflection on the way through but Cristian Alvarez readjusted well to make an important save.
Javi Martinez had Espanyol's first real sight of goal, striking left-footed from distance but not beating Gorka Iraizoz in the away goal.
Espanyol's opener came from their next opportunity, Alvara Vazquez crossing from the left and locating Garcia, who nodded home a low header at point-blank range.
Garcia attempted to double his tally soon after but this time could not find a way past Iraizoz from the edge of the area.
Bilbao rallied and enjoyed 10 minutes of heavy possession, during which time Alvarez was in action on more than once occasion.
Llorente had the best chance of that period, breaking through the centre and chipping towards the far corner only for Espanyol goalkeeper to claw the ball to safety with his trailing arm.
Gabilondo and Ander Herrera both shot tamely at Alvarez too, while Thievy Koulossa - an early replacement for the injured Juan Albin - almost went clear for Espanyol in first-half injury time.
Bilbao sent on both Markel Susaeta and Ander Iturraspe at half-time and the latter almost levelled within four minutes, volleying goalward from 12 yards only for Alvarez to make another crucial intervention.
Susaeta also made his presence felt, his 52nd-minute free-kick drawing another athletic stop from Alvarez.
Susaeta also provided the corner from which Llorente rose to head the equaliser, with Garcia almost doing enough to nod the ball off the line.
Espanyol regained the lead in the 74th minute when Garcia touched Joan Verdu's chipped pass towards goal, where Iraizoz blocked.
Garcia took one touch to take the loose ball high over the keeper and then showed great awareness to overhead kick over it the line from close range.
Llorente continued to threaten but Espanyol were able to close out the win without undue stress.