The rain hindered Connacht's running game and Munster hooker Rhys Marshall's second-half try was the crucial score.
Ian Keatley kicked Munster's other 11 points including two first-half penalties as they led 6-3 at the break.
Jack Carty slotted Connacht's three penalties as they failed to capitalise on several good attacking positions.
The key phase of the match was a two-minute period midway during the second half - as Marshall finished off a line-out maul to score after Connacht had squandered a chance of their own when Tom McCartney had been dispossessed following a poor pass from Denis Buckley.
As Munster broke quickly, Keatley's superb crosskick found a charging Andrew Conway and after Buckley had been bundled into touch near his own line, the visitors mauled their way to the crucial try.
Helped by a wind advantage, Connacht had the majority of possession in the first half but while Rory Parata, Matt Healy and Carty went close to scoring tries, the Munster line was not breached as the champions suffered their seventh defeat of the campaign.
Because of the IRFU's player management programme, Munster boss Rassie Erasmus had to leave out stars Conor Murray, CJ Stander, Simon Zebo, Donnacha Ryan and Peter O'Mahony as he made 11 changes from the Boxing Day win over Leinster.
The absence of in-form fly-half Tyler Bleyendaal might have looked an even bigger loss for Munster but his replacement Keatley went on to produce an impressive display for the visitors.
Connacht went into the contest with more than 20 members of their squad ruled out but Pat Lam was still able to parade a starting line-up that included Ireland internationals Tiernan O'Halloran, Matt Healy, Kieran Marmion, Finlay Bealham and Niyi Adeolokun.
After Keatley's early penalty, the Munster fly-half was nearly caught out as he attempted to shepherd a Carty grubber kick over his own line, with Marmion's attempt to get the touchdown only ruled out after a prolonged viewing by the TMO.
As Connacht failed to exploit their wind advantage and decided edge in the scrum, Munster led 6-3 at the break as Keatley restored his side's lead just before the interval.
Carty's levelling penalty on 50 minutes spurred Connacht to produce more pressure but the turnover which led to Munster's try proved to be the key moment of the match.
After Keatley's successful conversion, his further drop goal extended Munster's lead to 16-6 and while Carty's late penalty left only seven between the teams, Connacht had to settle for the losing bonus point as the sin-binning of Dave Kilcoyne and Duncan Williams came too late to upset the visitors.
Connacht coach Pat Lam: "There were just too many errors in possession. Dropped balls and turnovers.
"There was a lot of heart and the boys hung in there.
"The key moment of the game was (in the second half) when we were one metre from their line and we turned the ball over and they went down the other end to score. These big games come down to little margins."
Munster director of rugby Rassie Erasmus: "I'm very proud of the boys and they can be very proud of each other tonight.
"After losing the last few times against Connacht, to get a win against them is very satisfying.
"It was not a great performance because of the weather but because of the conditions, we can be very satisfied."
Connacht: T O'Halloran; N Adeolokun, R Parata, P Robb, M Healy; J Carty, K Marmion; D Buckley, T McCartney (capt), F Bealham, Q Roux, J Cannon, N Fox-Matamua, J Heenan, J Muldoon
Replacements: D Heffernan for McCartney 58, JP Cooney for Buckley 77, J Andress for Bealham 71, L Stevenson for Heenan 73, S O'Brien for Fox-Matamua 29, C Blade for Marmion 69, N Dawai for Roux 59
Not used: Gaffney
Munster: A Conway; K Earls, F Saili, D Goggin, R O'Mahony; I Keatley, D Williams; J Cronin, R Marshall, S Archer; J Kleyn, D Foley; B Holland (capt), T O'Donnell, J O'Donoghue.
Replacements: D Kilcoyne for Cronin 53, T Du Toit for Archer 49, R Copeland for O'Donnell 71, C Oliver for Foley 70.
Not used: J Taute, R Scannell,  T Aihe Toma, K O'Byrne
Referee: Dudley Phillips (Ireland)
Munster extended their lead at the top of the Pro12 to five points as they beat defending champions Connacht in a battle of attrition in a wet Galway.