A bright start from West Brom saw Rondon cause problems with his physicality, and Morrison headed just wide of the Stoke post in the fifth minute as the visitors were forced onto the back foot in the early exchanges.
Stoke had a half-chance of their own in the ninth minute when Ryan Shawcross found fellow centre-back Bruno Martins Indi inside the West Brom box, but the summer arrival could only poke his effort wide of the post.
Kurt Zouma was next to come close for the improving visitors in the 24th minute after the centre-back's long-range effort looped towards the West Brom goal, and Baggies goalkeeper Ben Foster had to be alert to tip over the crossbar.
Jack Butland was called into action down the other end as he turned a deflected Craig Dawson effort wide of the post 10 minutes before the break, but a fairly drab first 45 minutes at The Hawthorns finished goalless.
It was a similar story at the start of the second period, with the opening 15 minutes of the half passing without incident.
The hosts should have doubled their lead in the 66th minute when Ahmed Hegazi met a free kick from Chris Brunt, but Butland was on hand to make a smart save, before Rodriguez dropped a header on the top of the crossbar in the 73rd minute.
Stoke levelled the scores in the 77th minute, however, when substitute Crouch headed home from close range after a mix-up in the West Brom box between Hegazi and Foster.
Zouma came close to sending Stoke ahead in the 85th minute with a powerful effort from distance, but the centre-back's strike missed the post, before Choupo-Moting fizzed one just wide two minutes later as the points were shared following a largely uneventful 90 minutes of football.