A first goal for the club from Jay Rodriguez just past the hour mark had looked enough to settle a low-quality encounter before Crouch levelled the scores 13 minutes from time.
West Brom won each of last season's league meetings 1-0 and the latest showdown showed few signs of being a more goal-laden affair, with neither side able to muster a clear chance in a tepid first 45 minutes.
Stoke, which performed so impressively to beat Arsenal last time out, enjoyed more of the ball in the West Midlands sunshine but were unable to create any real danger against a Baggies side that had kept clean sheets in each of their first two games.
It looked as though Rodriguez's brave header would be enough to give Pulis a fifth win in six games against Stoke since taking charge of West Brom, but a dreadful mix-up between Ahmed Hegazi and Ben Foster gifted Crouch the equaliser.
Stoke is still yet to beat West Brom since their former boss Pulis arrived at The Hawthorns, but the draw was enough to lift it above Everton into ninth in the table and deny the home side the chance to move level on points with leaders Manchester United.
Kurt Zouma was the unlikely figure to come closest to breaking the deadlock twice in the opening half-hour, the centre-back heading over from a corner before seeing a deflected shot tipped over the crossbar by the alert Ben Foster.
West Brom was content to allow Stoke the majority of the possession but Mark Hughes' side showed few signs of making the most of it, with Geoff Cameron's inexplicably poor strike from a good position inside the penalty area summing up the dearth of first-half attacking quality.
Just one shot on target from either side in the first hour suggested that an encounter typically low on goals would be unlikely to buck the trend but, with 61 minutes played, West Brom at last gave the home fans something to shout about.
Allan Nyom got forward once more down the right and this time produced a fine deep cross, allowing Rodriguez to stoop down and head the ball beyond Jack Butland from six yards.
Butland made a terrific reaction save to repel Hegazi's header moments later as West Brom pushed for a second, with Stoke unable to keep the ball long enough to provide service to substitute Crouch.
But the host undid its good work 13 minutes from time, as Hegazi and Foster both went to clear a cross from Ramadan Sobhi and failed to connect with the ball, allowing Crouch the simple task of nodding it into the empty net.
Zouma dragged an effort narrowly past the left-hand post and Eric Maxim Choupo-Moting scuffed a late shot wide, but Stoke were unable to find what would have been a first victory against West Brom since December 2014.