The opening goal for Albion came down the right-flank, Rodriguez met a cross from Allan Nyom and finished past Jack Butland.
While Stoke equalised following an uncharacteristic error from a Tony Pulis team, as Ahmed Hegazi and Ben Foster both went for a cross, neither really claimed it and Crouch finished with ease.
Drab first half
Stoke started the game in a 3-4-3, but with Geoff Cameron in at wing-back instead of Mame Biram Diouf. The away side lined up in a very rigid shape, and Mark Hughes appears to be another manager shoehorning players into a 3-4-3 shape even though they may not fit it.
The first chance of the game fell to Bruno Martins Indi following a corner that Ryan Shawcross turned goalwards, his flick dribbled just wide of Ben Foster’s goal though.
Another good set-piece routine for Stoke saw Kurt Zouma nod over from a few yards out after contesting with West Brom’s new cult hero Ahmed Hegazi.
Zouma then had another shot from 20 yards deflected high and almost found its way in the back of the net off Garry Barry if not for Foster’s quick feet.
Despite some late pressure in the half by both sides, neither were able to fashion any real chances before half-time.
Second half filled with action
Pulis is unbeaten in all six Premier League managers against Stoke since he left the club, and despite the second half not continuing in the same vein as the first, that run was never in doubt of coming to an end.
Albion started brightly, and had the first chance of the second half, Rodriguez sent his side ahead with his first goal for the club.
Nyom did well down the right-flank, and put a superb cross in towards the back post where Rodriguez met with a header ahead of Zouma.
Hegazi then nearly doubled Albion’s lead, meeting Matt Phillips' free kick but his header was tipped over by Butland.
Stoke then equalised through substitute Crouch, after a lapse in communication between Hegazi and Foster allowed the striker to finish with ease.
The goal is Crouch’s 51st headed strike, the most by any Premier League record. A record he has held for a while now.
Joe Allen then combined with substitute Ramadan Sobhi on the right, before his shot was palmed away by Foster as both teams pushed for a late winner.
Eric Maxim Choupo-Moting then nearly won the game for Stoke, bringing down a deep cross well before his strike was superbly blocked by Nyom.
Salomon Rondon then skewed his shot wide in stoppage time, as the game fizzled out to a draw.
The result leaves West Brom on seven points to start the season, while Stoke go into the international break on four points from three games.
While both managers will be relatively happy with a point, they will also feel that this was an opportunity for three points wasted.