Mubarak Wakaso opened the scoring for the hosts on the stroke of half-time, but Sergio Canales levelled shortly after the re-start.
Joan Verdu then put Espanyol ahead again late on but his effort was cancelled out by Jonas, before Roberto Soldado gave Valencia the lead for the first time in the game, during injury time.
However, there was still time for more drama, and Garcia scored in the 94th minute to snatch the draw.
The game took a while to kick into action as Valencia failed to impose themselves on their hosts, who had more of the ball but could not do much with it.
Espanyol thought they should have been awarded a penalty after 15 minutes when Christian Stuani was brought down by Andres Guardado but the referee waved play on.
The first proper chance of the game fell to Garcia, who was through on goal but shot just wide of the far post. A minute later Valencia goalkeeper Vicente Guaita kept out a shot by Stuani at his near post, and before the end of the half got down low to save a powerful shot from Joan Capdevila.
Wakaso gave the hosts the lead their pressure deserved right on half-time, arriving at the back post to head in a cross from Garcia.
Valencia made one change at the break, bringing on veteran David Albelda in place of Tino Costa, and they were a much improved outfit in the second half.
They levelled eight minutes after the re-start, Canales getting on the end of a ball from Ever Banega before shooting through the legs of Kiko Casilla.
Soldado came close to giving his side the lead when he hit the post after shooting on the turn following a corner, and the striker missed a golden opportunity on 79 minutes, hitting a soft shot straight at Casilla from 10 yards out.
Soldado's miss preceded a final frantic 10 minutes which saw four goals.
First Verdu restored Espanyol's lead following good work by Wakaso and Garcia but Jonas equalised with three minutes remaining, finishing off a fine move. Victor Ruiz pumped a long ball into the box and Soldado laid it off for Jonas, who slotted into the bottom corner.
There was still more action to come.
Raul Baena thought he had won the match for Espanyol but his goal was ruled offside after it deflected off one of his team-mates on its way into the goal.
Soldado then scored what looked certain to be the winner in the 91st minute, getting in behind the defence and shooting into the bottom corner to make amends for his earlier misses.
The Catalans looked to be heading for their first home defeat under coach Javier Aguirre but Garcia equalised with the last kick of the game, shooting into the roof of the corner from a tight angle in the six-yard box.
The result means Espanyol overtake Levante in 11th place in the La Liga table, while Valencia missed a chance to provisionally reclaim fourth spot from Real Sociedad, who play Rayo Vallecano on Sunday.