The capital club lost all six of their pool games last season but opened up this year's campaign with their first success since beating Toulouse in the quarter-finals of the 2011-12 competition.
Matt Scott had given the home side the lead just six minutes in but tries from Casey Laulala and Mike Sherry handed the advantage to Munster.
Greg Laidlaw was his ever reliable self with the boot and four penalties before the break had Edinburgh ahead again, only for opposition kicker Ian Keatley to strike twice himself after the interval as the Irishmen edged themselves ahead.
But with 10 minutes left and Munster growing increasingly confident, Visser sprinted down the left-hand wing to to grab the try which set up a unlikely victory, sealed with another two successful kicks from captain Laidlaw.
Five defeats out of six in the RaboDirect Pro 12 had done nothing to dilute the air of expectancy in the capital as a 7,000-strong crowd turned out to see Edinburgh kick-off their European campaign.
Their slow start has been blamed on a combination of a hangover from last year's miserable efforts - just seven wins in the league and none at all in the Heineken - and the late arrival of new boss Alan Solomons just a month before the start of the season.
But there were early signs that the South African was finally beginning to make his mark as the hosts piled on the pressure on their esteemed opponents in the opening stages.
That build-up paid off just six minutes in as Scott took advantage of a three-on-two overload on the left to dart in for the opening try, with Laidlaw adding the conversion with the assistance of the far upright.
The seven-point advantage soon became 10 as Laidlaw stuck over a penalty but some clever interplay with British and Irish Lion Conor Murray and winger Simon Zebo teed up Keatley to put Munster on the scoreboard with a penalty of their own.
Parity was restored 20 minutes in as Laidlaw's kick was charged down before Munster found Laulala in scoring territory from two yards, with Keatley adding the extra two.
Another Laidlaw penalty had the hosts back in front but Munster continued to pour forward, only a forward pass denying prop Stephen Archer from adding the visitors' second score.
Edinburgh were having to defend ever deeper and that handed hooker Sherry the opportunity to barge himself through a ruck to put Munster ahead for the first time on 27 minutes before Keatley struck again with the conversion.
Yet the hosts remained competitive, mostly thanks to Laidlaw's flawless kicking, with the skipper adding another two penalties from out wide to see the Gunners back in front 19-17 at the break.
Munster's Kiwi head coach Rob Penney has abandoned the pick-and-go style that twice took the Red Army to Heineken success in the last seven years and replaced it was a more patient approach.
That new style was evident as they waited for an Edinburgh mistake before regaining the lead 13 minutes into the second period, with Keatley punishing the home team's Grant Gilchrist for holding on in the tackle with another penalty.
Edinburgh, though, were happy to scrap but their enthusiasm got the better of them again as handling at the breakdown gave Keatley the chance to kick Munster's lead to four points with 15 minutes remaining.
But just as Munster looked ready to go up another gear, Visser collected from Gilchrist to run in from 30 yards for a sucker punch the visitors did not see coming. The TMO was called in to rule on Scott's collision with JJ Hanrahan as the Munster replacement let fly with the boot in the build up but the Scotland centre was cleared of obstruction as the home fans celebrated.
Laidlaw again added the extra points to give Edinburgh a three-point lead before he knocked over a fifth penalty to take him beyond 500 points for the capital club, clinching a memorable victory in the process.
- Sports & Recreation
- Ian Keatley