It was the hosts who were more creative in the final third, but their first chance was handed to them when Grant Hanley's tame back pass was intercepted by Chris Brown, who poked just wide of the post.
After David Cotterill and Richie Wellens had tested Jake Kean, Robinson, who only signed from Derby County on Thursday, beat the Blackburn stopper on 37 minutes with a near-post header from James Husband's free kick.
Chris Taylor saw what he thought was an equaliser ruled out just before the break when Rhodes was adjudged offside, but few more opportunities came the visitors' way during the remainder of the contest.
Instead, Donny extended their advantage as provider became scorer after the interval, when Husband's attempted cross looped over the backtracking Kean and dipped into the far side of the net.
Gary Bowyer's Blackburn are now without a win in their opening three Championship matches of the campaign.