McDowell posted a four-under-par 68, including six birdies and two bogeys, to lie in a tie for 11th place on eight under par after day three.
Earlier on Saturday, McDowell completed his second round 72, Friday's action having been abandoned midway through the day because of strong winds.
Darren Clarke missed the cut on three over, while Paul Dunne lies one over.
Garcia carded a third-round 68 to enjoy a three-shot lead over fellow Europe Ryder Cup player Henrik Stenson, who shot a 67.
Ian Poulter and Prom Meesawat are joint third on 11 under.
Graeme McDowell trails leader Sergio Garcia by eight shots after the third round of the Dubai Desert Classic.