The home side enjoyed almost all the territory in the opening quarter and had their just reward in the 15th minute when, from a scrum after JJ Hanrahan had knocked-on, prop Rodney Ah You crashed over close to the uprights to leave the conversion a formality for Matt Jarvis.
Hanrahan made amends for his earlier error when he landed a penalty into the wind minutes later and he further cut the deficit with a 25th minute penalty as his side began to enjoy their share of possession and position.
Jarvis had the opportunity before half time to kick his side in front but pushed his penalty wide and the out-half ended the half watching from the bench after been yellow carded for preventing release.
A further penalty allowed Hanrahan kick Munster to five metres from Connacht's line but the lineout throw was over-cooked and the chance went a-begging.
Connacht were further reduced when Ethienne Reynecke saw yellow but from the subsequent scrum Munster were unable to gain advantage and the sides went into the break with the home side leading by a solitary point.
With the wind at their backs Munster opened the second half brightly, Ronan O'Mahony making good headway down the left wing supported by Hanrahan, Connacht conceded the penalty but Hanrahan's effort tailed left and just wide.
Although playing into a stiffening wind Connacht took the game to the visitors who needed to be alert to deal with the lively wing Steve Macauleyand the probing drives of TJ Anderson and Mick Kearney with Anderson just failing to send Kearney clear after his clean break on half way in the 54th minute.
Anderson and Kearney were again prominent as Connacht forced the pace that saw them held up over the line, Munster shredded the subsequent scrum but handed back possession and a foot in touch denied the home side their second try.
Having been confined to the environs of the 22 for the best part of half an hour, Munster eventually lifted the siege but their sojourn in Connacht territory was brief and they found themselves back inside their 22 after being were penalised at a scrum just inside their half for Jarvis to find touch five metres out.
Jarvis kicked Connacht four points clear just before the end of normal time, then left Munster requiring a converted try to draw when he landed another in added time.
To their credit Munster ended the game laying siege to the Connacht line but the home side held on for a deserved victory.
Connacht Eagles: S Layden; S MacAuley, T Leader, D McSharry, C Finn; M Jarvis, M Walsh; F Bealham, E Reynecke, R Ah You; M Kearney, D Qualter; TJ Anderson. E Grace, J Gardiner. Replacements: K O'Byrne, P Reilly, J Walshe, D Heffernan, A Conneely, B Digby, D Leader, F Gormley, A O'Meara.
Munster A: L O'Dea; R O'Mahony, I Dineen, S Scanlon, D Barnes; JJ Hanrahan, N Cronin; D Kilcoyne, D Casey, J Ryan; P Donnellan, D O'Shea; S Buckley, B O'Mahony, S Rennison. Replacements: J Cronin, G Slattery, C Condon, Ed Kelly, J Murphy, J Ryan, J Holland, C Hircock, C Bohane, R Mullane.