The first half was dominated by the boot of Lee Thomas and Paul Burke - each kicking three penalties to leave the scores tied at 6-6 at the break.
The second period saw Munster display more willingness to throw the ball around, but it still required two more Burke penalties to keep the scoreboard moving.
After the second, Burke departed, to be replaced by Shaun Payne, with full-back Jeremy Manning moving to number 10 and taking over the kicking duties to further extend the visitors' lead.
Even with Italian prop Federico Pucciariello in the sin-bin, Munster retained control, with the Blues hammering in vain at the Irish defence. As soon as he returned, the home side broke through, a fine handling move ending with Australian centre Marc Stcherbina darting over. Thomas converted from wide out.
But even though the Welsh region threw the ball around in desperation, Munster kept them penned in their own half to secure a deserved if narrow victory.