Munster held on to win 16-10 in a frantic finish in Galway as they dashed Connacht's hopes of building on last week's shock Pro14 win over Leinster.
Chris Farrell's 19th-minute try helped Munster lead 10-0 as home hooker Shane Delahunt was in the sin-bin.
Jack Carty slotted a Connacht penalty before half-time but JJ Hanrahan's two penalties put Munster 16-3 ahead.
But two late Munster yellow cards saw Peter Sullivan score a Connacht try and the home side almost notched another.
Ultan Dillane was held up just short of the Munster line in the fourth minute of injury-time after Nick McCarthy had joined Rory Scannell in receiving a yellow card.
With Munster conceding a penalty, Connacht opted for a five-metre scrum against the 13 men but a knock on saw the relieved visitors hang on for victory.
Try-scorer Farrell was among 11 internationals who returned for Munster after their second-string side's narrow defeat by Ulster in Belfast.
Prior to the dramatic finale, Connacht had produced four periods of intense pressure in the Munster 22 but they were turned over on each occasion by the visitors' disciplined defence.
In the first half, despite a heavy pitch, Munster were able to produced moments of penetration - most notably Ireland centre Farrell's try as he finished a line-out maul created by the visitors winning a penalty in their own half after turning over Connacht centre Sammy Arnold.
Farrell's try came after home hooker Delahunt had been yellow carded in the ninth minute for pulling down the maul which led to Hanrahan's first successful penalty.
Other frontline players who returned to the Munster included Farrell's fellow Ireland internationals Conor Murray, Keith Earls, Tadhg Beirne, Peter O'Mahony and CJ Stander plus South Africa's World Cup winner Damian de Allende.
The maul was a particularly potent weapon for Munster in the opening half which set the tone for the majority of the contest.
Despite Carty opening Connacht's account before half-time, Munster quickly re-established a 10-point advantage as Hanrahan punished Tom Daly's offside before the fly-half increased the margin to a seemingly safe 13 points in the 62nd minute.
While Connacht continued to battle, the contest seemed to be petering out before suddenly the yellow carding of Munster replacement Scannell on the edge of his own 22 in the 77th minute gave the home side a late lifeline.
With Munster short on numbers, replacement Sullivan barged his way over with Carty slotting a brilliant conversion from the touchline after an earlier inexplicable penalty miss in the first half when the match was scoreless.
After winning possession from the restart, Connacht chased a dramatic victory but despite forcing two penalties as Dillane was held up inches short, they botched a five-metre scrum in the final play as Munster held on.
Munster's win increases their advantage over their Irish rivals at the top of Conference B to 11 points and Johann van Graan's side remain odds on to clinch a place in the Pro14 Final
Connacht: Porch, O'Donnell; Arnold, T Daly; Wootton; Carty, Blade; Bealham, Buckley, Delahunt; Dillane, Roux; O'Brien, Oliver, Masterson.
Replacements: Heffernan, Burke, Robertson-McCoy, Thornbury, Boyle, Marion, Kilgallen, Sullivan.
Munster: Haley; Earls, Farrell, de Allende, Daly; Hanrahan, Murray; Cronin, O'Byrne, Archer; Kleyn, Beirne; Coombes, O'Mahony (c), Stander.
Replacements: N Scannell, J Wycherley, Knox, F Wycherley, Holland, McCarthy, Healy, R Scannell.