Cronin made the most of the one man advantage after Lee Jones saw yellow after 25 minutes, by scoring under the posts, but Edinburgh responded after the break with a Nick De Luca try.
Ivan Dineen grabbed another Munster try a few minutes later and JJ Hanrahan extended the lead before Jones pulled one back for Edinburgh but the hosts held on.
Munster started brightly and Sean Dougall almost finished an excellent move in the first five minutes but knocked on when closing in on the try line.
And it was the visitors who opened the scoring when Francis snuck a penalty just inside the right hand post to give Edinburgh the lead.
Munster hit back immediately as Ian Keatley found Hurley on the wing and he touched down in the corner to put the hosts ahead.
And captain Coughlan extended their advantage after a cheeky kick from Johne Murphy bounced perfectly for the No.8 and he crossed before Keatley converted.
Keatley then added three more points from a simple penalty right in front of the posts to put Munster ahead 15-3 after 20 minutes.
Francis pulled Edinburgh to within nine points with another penalty five minutes later but Jones was yellow carded shortly after for tackling Coughlan in the air.
And Munster took advantage when Cronin, who was promoted from the academy this summer, powered over under the posts and Keatley slotted an easy conversion.
The Edinburgh pack continued to be their most effective outlet and they earned another penalty just after the half hour mark but Francis could not make it count.
But they did get themselves back in the game ten minutes into the second half when De Luca darted through the Munster line to score under the posts before Francis converted easily.
Muster responded immediately, and secured the bonus point with their fourth try, when Dineen finished an excellent move, crossing the whitewash on the wing.
Harry Leonard replaced Francis just after the hour mark and immediately reduced the Scottish side's deficient by three-points with a central kick.
But replacement pair Cathal Sheridan and Hanrahan combined well from a Munster scrum with the latter crossing under the posts before Edinburgh grabbed a consolation try through Jones in the final ten minutes.
|Munster Rugby Score Card|
|Edinburgh Rugby Score Card|
|Nick de Luca||1||5|