Tony McGahan’s men went into this one with 14 straight wins under their belts but they never got on top and were always playing catch-up to an Edinburgh side whose only other win this term was against Connacht.
Bradley has now led the Scots to victory over his native province and the Connacht side he coached for seven years.
It was a disappointing outcome for the champions and leaders but it was Edinburgh who dictated the play from the early stages and they did not allow Munster to settle in this entertaining RaboDirect PRO12 clash.
Edinburgh, anxious to bounce back from their defeat in Aironi last week, took the game to Munster from the outset and raced into 16-0 lead after just 21 minutes.
The Reds were on the back foot from the outset and Greig Laidlaw settled Edinburgh with a penalty after four minutes from the left after Scott Deasy had been penalised for holding on.
He added another three minutes later after Tim Visser had made good yardage, and the scrum-half made it 9-0 after 15 minutes with a simple kick.
Munster were forced to do plenty of defending in that opening quarter and the pressure eventually paid off for Edinburgh when Matt Scott, starting just his second match, crowned his 21st birthday with an excellent try when he burst through the middle from outside the 22 for a score which won’t make for pleasant viewing during the visitors’ video session on Monday.
Laidlaw added the points to make it 16-0 and there was a grave danger that the game would just slip away from Munster.
However, they regrouped and started to build the phases and Ian Keatley got them off the mark after 26 minutes with a penalty from the right wing.
He was off target with a drop goal attempt four minutes later after another sustained bout of pressure from McGahan’s men but Edinburgh were forced to concede penalties as Munster started to enjoy sustained possession in the home side’s half.
Keatley reduced the margin with another penalty from the right from about 37 metres and left it 16-9 at the interval with one from the opposite side from 22 metres after Edinburgh were penalised in the scrum.
But it was Edinburgh who seized control after the restart with Laidlaw increasing their lead with a 44th-minute penalty, this time from 30 metres.
Munster countered but Edinburgh replacement prop Shaun Knight, on loan for six weeks from Gloucester, pounced on his own 22 to turn over possession, with Gregor Hunter making ground with an audacious chip and short chase and they finished the move at the other end with Laidlaw and Lee Jones combining to send hooker Steve Lawrie over.
Laidlaw added the points to make it 26-9 to Edinburgh after 47 minutes, leaving Munster with a mountain to climb.
Invariably, they battled back but Edinburgh defended with discipline and never gave Munster a chance to eat into the lead, with the superb Dave Denton leading the home charge from the base of the scrum.
Danny Barnes gave Munster notions of a losing bonus point when he got his second try of the campaign eight minutes from the end when he squeezed over in the right corner after a final pass along the line by Denis Hurley.
But it was Edinburgh who again hit back, with Laidlaw making it seven from seven with the boot as he landed another penalty to complete a fine win for Bradley’s men.
Scorers for Edinburgh: Tries: Scott, Lawrie. Cons: Laidlaw 2. Pens: Laidlaw 5.
Scorers for Munster: Try: Barnes. Pens: Keatley 3.
EDINBURGH: Thompson, Jones, Scott, King, Visser, Hunter, Laidlaw, Traynor, Lawrie, Niven, Gilchrist, Cox, Denton, Grant, McInally.
Replacements: Hislop for Traynor (63), Walker for Lawrie (70), Knight for Niven (39), Lozada for Gilchrist (61).
MUNSTER: Deasy, Murphy, Gleeson, Mafi, D. Hurley, Keatley, Stringer, du Preez, Fogarty, Hayes, Foley, O’Driscoll, O’Donnell, Ronan, Butler.
Replacements: Barnes for Gleeson (50), Zebo for Keatley (61), Williams for Stringer (73), J. Ryan for du Preez (75), Sherry for Fogarty (50), Archer for Hayes (50), Nagle for Foley (61), O’Mahony for Butler (50).
Referee: Leighton Hodges (RFU).