And those misses would prove costly with Munster fly-half Ian Keatley finishing with an 18-point haul, including one try, with wing Luke O'Dea also crossing early in the second half.
Visser broke the game's deadlock in classic style in the 13th minute as he was brilliantly set up by debutant Greig Tonks.
The full-back emerged out of the shadow of centre Nick De Luca and drew his man to put Visser in however Laidlaw missed the conversion.
Munster's pack predictably led the fight forward and they won a penalty at the scrum on 22 minutes which Keatley snatched at. But the 25-year-old atoned five minutes later after Edinburgh's hands were all over the ball at ruck time.
Visser then found some space on his wing after Edinburgh's fearsome back-row turned the ball over and he fended off the last man to go over unopposed.
But Munster's new captain Doug Howlett matched up to his opposite man by creating a try to level things up at half-time.
The 33-year-old former All Black chipped over the home side's advancing defence, the covering Laidlaw looked to sweep the ball up but the bounce evaded him and it was Keatley who benefitted.
In converting his own try it was 10-10 at the break and from there Munster barely looked back.
The collective arm wrestle between the sides in the first ten minutes after the break was broken only by Keatley's penalty which was then matched by one from Laidlaw after intense pressure from the home side.
Edinburgh tried to get the ball into Visser's hands as much as possible but increasingly the newly capped Scotland international found himself double and even triple marked at times.
The Irish province pursued the direct approach in riposte and it paid off with a try for O'Dea.
Powerhouse centre James Downey, who returned to the province from Premiership-side Northampton in the summer, showed a nice sleight of hand which put the 22-year-old O'Dea in.
Downey latched onto a lineout on Edinburgh's 22, Keatley fed him into midfield and O'Dea was on his shoulder to go over right next to the posts.
Keatley converted before adding another penalty as Munster seemed to squeeze the life out of Edinburgh.
Yet they couldn't quite keep a grasp on Visser and with six minutes remaining Laidlaw moved play to the left wing and Denton passed on to De Luca who darted through a hole before offloading to the Dutch-born speedster.
But once again Laidlaw failed to convert and Munster just managed to shut them out for an opening victory to their campaign.
|Edinburgh Rugby Score Card|
|Munster Rugby Score Card|