Anthony Foley's men triumphed in Cork to move back into the Guinness PRO12's top six, going five points clear of the Scots, who have fallen to seventh place.
The hosts led 20-13 at half-time thanks to tries from Rory Scannell, Simon Zebo and Andrew Conway - all in the same right corner - with Jason Tovey responding with two penalties and a superb solo seven-pointer.
Penalties from Tovey and replacement Sam Hidalgo-Clyne were the only scores of a nervy second half until centre Saili was driven over for the clinching touchdown.
Edinburgh had the better of the early exchanges at Irish Independent Park, pressing from an advancing maul, with scrum-half Sean Kennedy linking crisply across the back-line.
But the men in red hit the front in the sixth minute, with Tommy O'Donnell leading the charge off a couple of turnovers and man-of-the-match Conway's neat one-handed offload putting centre Scannell over out wide.
The difficult conversion was expertly negotiated by Johnny Holland, whose opposite number Tovey nailed a left-sided penalty on the quarter hour, following up on an excellent break by Matt Scott.
Tovey quietened the home crowd further with a peach of a try in the 18th minute, gliding in between two front-rowers and embarrassing Zebo with a final step to score to the left of the posts and convert.
A well-struck Holland penalty was followed by a poor Tovey miss, and Munster stepped up a gear when Conway squeezed his way through and chipped from halfway for the onrushing Zebo to collect and touch down.
After Holland missed the conversion, he spearheaded a promising left-wing attack involving Keith Earls.
CJ Stander was then held up past the try-line before some good phase-building led to Conway getting over for a deserved score, with Stander and Billy Holland combining well in the build-up.
As the rain came down, Tovey was back on target with a late penalty to make it a seven-point game.
Tovey opened the second-half's scoring with his third successful penalty to reduce the deficit to 20-16, as Munster's streak of indiscipline allowed him to chip away at the lead.
Their WP Nel-inspired scrum gave Edinburgh a reliable platform and the game turned into a real dogfight, with individual mistakes preventing Munster from building any attacking momentum.
Another scrum penalty saw Hidalgo-Clyne brilliantly split the posts from long range, reducing the margin to just one with a quarter-of-an-hour remaining.
Munster replacement Mike Sherry was hauled down just short after a storming 30-metre maul, but two awful kicking errors from Conor Murray and Zebo allow the Scots to come right back at them.
Finally turning the screw, Foley's side went for touch off a Matt Scott penalty, going close again via the maul, with an Earls effort ruled 'inconclusive' by TMO Brian MacNeice.
The Munster maul proved to be a real weapon in the closing stages, though, gaining some crucial yards, and another bit of forward power took Saili in under the posts to leave Edinburgh floored, with Holland's simple conversion taking away their losing bonus point.