Baggies head coach Tony Pulis selected a strong starting lineup at Plainmoor with the likes of Ben Foster, James Chester, Jonny Evans, Craig Gardner and Salomon Rondon all being named in the team but they struggled to create chances during the opening quarter.
Gardner fired off-target after receiving a pass from James McClean, while Rondon squandered an opportunity to get a shot away after being found in the final third.
At half time, Pulis made a number of substitutions - which included the introduction of Saido Berahino - but seven minutes after the restart, Torquay went in front after Callum Hall found the corner of the net from 25 yards.
Dawson almost made amends when his header clipped the crossbar before Matt Phillips also struck the woodwork from distance but Torquay reached the final 20 minutes of the game with a notable scalp within their reach.
With nine minutes left, however, West Brom finally made their pressure count when Phillips produced a classy finish after being found by a pass from Darren Fletcher and the goal saw the Baggies lay siege to the Torquay goal during the closing stages.
That proved to be the last chance of note as the non-League outfit held out for an impressive win just five days before they start the new season away at Macclesfield Town.