Munster reclaim top spot
It looked as though Munster's winning sequence would be ended as Jason Tovey scored three first half penalties to hand the Scots a half-time lead.
However, Munster had remained in the contest, scoring their points thanks to a try from Ronan O'Mahony and a conversion from Bleyendaal.
But after a scoreless second half, Munster claimed the narrowest of wins against their Scottish opponents, with Bleyendaal scoring the crucial penalty in the final quarter of the game.
Edinburgh scored the first points on seven minutes, with flyhalf Tovey firing over a penalty after Munster had been penalised at scrum-time for wheeling the set piece.
The visitors had targeted scrum dominance before kick-off, with Edinburgh minus a host of front-row options, but it was they that struck the first blow at the set piece at Myreside.
Munster are the form side in the Pro12 presently having won their last six games, with three of those wins earning a try-bonus point since their 14-25 reversal to Leinster at the Aviva Stadium on October 8.
But it was Edinburgh who reacted best in the early stages to missing a host of star names to this weekend's Six Nations as they dominated territory and possession.
The Irish province recovered after a slow start and wing O'Mahony was unlucky not to finish a dynamic break and kick through on 13 minutes with a score but Edinburgh's defence managed to recover and touch the ball down behind their line.
But it didn't take long for Munster to score the game's first try as Bleyendaal created the opportunity with a smart off-load just metres from the line on 15 minutes and O'Mahony rumbled over from close range. The flyhalf added the extras to make it 7-3.
The game turned on its head just ten minutes later as Munster scrumhalf Duncan Williams was sin-binned for kicking the ball out of his opposite number Sean Kennedy's hands and Edinburgh capitalised.
Tovey kicked the resulting penalty to bring his side back within a point of the high-flying visitors but the home side struggled to make their numerical advantage count in the ensuing ten minutes as Bleyendaal continually kicked for territory.
Edinburgh got their noses in front for the first time on the night, five minutes before the break, as Tovey slotted home his third penalty after Munster, who have lost only one of their last ten encounters with Duncan Hodge's side, were once again penalised at the scrum.
Edinburgh ended a three game losing run in the Pro12 with their 24-19 victory against Zebre in Parma on New Year's Eve, but they looked a completely different side against Munster as they went toe-to-toe with the high-flying Irish side early in the second half.
In worsening conditions the game descended into a kicking duel after the break, with Kennedy and Edinburgh outdoing the visitors.
Jack O'Donoghue made a meaningful break just before the hour mark for Munster, showing explosive pace but after storming into the Edinburgh 22, the ball was turned over at the breakdown.
Munster got their noses back in front on 67 minutes with the first points of the second half. Edinburgh were penalised at the breakdown and Bleyendaal made no mistake with a penalty.
Blair Kinghorn was handed a late chance to win it for Edinburgh with a penalty from the halfway line with the final play of the game but his effort fell agonisingly short.
Munster dominated possession in the final ten minutes and saw the game out convincingly to keep their impressive Pro12 run of results going, notching their sixth away-day win of the competition this season.
The scorers:
For Edinburgh:
Pens: Tovey 3
For Munster:
Try: O’Mahony
Con: Bleyendaal
Pen: Bleyendaal
Teams:
Edinburgh: 15 Blair Kinghorn, 14 Damien Hoyland, 13 Chris Dean, 12 Phil Burleigh, 11 Tom Brown, 10 Jason Tovey, 9 Sean Kennedy, 8 Cornell Du Preez, 7 John Hardie, 6 Magnus Bradbury, 5 Ben Toolis, 4 Fraser McKenzie, 3 Murray McCallum, 2 Neil Cochrane (captain), 1 Jack Cosgrove.
Replacements: 16 Stuart McInally, 17 Derrick Appiah, 18 Nick Beavon, 19 Lewis Carmichael, 20 Viliame Mata, 21 Sam Hidalgo-Clyne, 22 Michael Allen, 23 Rory Scholes.
Munster: 15 Jaco Taute, 14 Calvin Nash, 13 Francis Saili, 12 Dan Goggin, 11 Ronan O’Mahony, 10 Tyler Bleyendaal (captain), 9 Duncan Williams, 8 Jack O’Donoghue, 7 Conor Oliver, 6 Dave O’Callaghan, 5 Dave Foley, 4 Jean Kleyn, 3 Stephen Archer, 2 Rhys Marshall, 1 Peter McCabe.
Replacements: 16 Kevin O’Byrne, 17 Dave Kilcoyne, 18 Brian Scott, 19 Darren O’Shea, 20 Billy Holland, 21 Tommy O’Donnell, 22 Angus Lloyd, 23 Rory Scannell.
Referee: Ben Whitehouse (Wales)
Assistant Referees: Sean Brickell (Wales), Dunx McClement (Scotland)
TMO: Neil Hennessy (Wales)