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Garcia, 33, took his tally for the season up to 13 with his efforts either side of the break at the Mestalla.
"We are a group of friends who try to fight with everything we have. Being in Europe is the goal," Garcia said.
Voro took charge for the first time of Valencia after Albert Celades was sacked on Monday after a loss to local rivals Villarreal.
The hosts drop to 10th five points behind Villarreal in sixth spot and the final Europa League berth with both sides having three games of the La Liga campaign to play.
Garcia, who joined the Basque club in 2015, opened his account after 13 minutes slotting home a low Inaki Williams cross.
He added a second goal two minutes into the second half with a long-range shot which fired past Valencia goalkeeper Jasper Cillessen into the top corner.
