Ritchie gave Swindon the perfect start by opening the scoring with only eight minutes gone but an own-goal from Danny Hollands two minutes later handed Doncaster point.
The game got off to a frantic start and Ritchie swept the hosts into the lead with a sweetly struck right-footed shot after Andy Williams had picked out the winger at the back post.
Swindon's joy was shortlived and Doncaster drew level when Hollands stretched to intercept a cross and could only turn the ball past home goalkeeper Wes Foderingham.
Chris Martin came close to restoring Swindon's lead before half-time when he headed a Rafaele De Vita cross against the post.
Williams also saw his shot rebound off the crossbar in the final minute of the half.
Swindon continued on the front foot after the interval and Ritchie, Jon Bostock and Williams all went close but Doncaster held on for a point they scarcely deserved.