Munster trailed 7-0 after just 11 minutes when Niall O'Brien scythed over for a try that Paul Warwick converted but that heralded a response from Munster that yielded three tries, from Anthony Horgan, Denis Leamy and Shaun Payne
Just before half-time a Munster error let Conor McPhillips in for a try but Munster went further ahead when the outstanding Christian Cullen scored just before the unterval after a superb Jason Holland break.
Back came Connacht with a Warwick penalty and an Adrian Clarke try and in the 65th minute Warwick's penalty looked good enough to give Connacht their first win over Munster since the mid eighties.
However, Munster kept plugging away, were unlucky perhaps not to score a fifth try but in the end will have been happy enough to get out with three points.
Munster; C Cullen; J Kelly, S Payne, J Holland (Henderson), A Horgan; P Burke, M Prendergast; F Roche, J Flannery, G McIlwham; T Bowman (Ryan), T Hogan; S Keogh, D Leamy, J Williams (capt).