The visiting side managed to edge ahead courtesy of two Ian Keatley penalties inside the opening 12 minutes, although the Vulcans soon hit back through captain Damien Chouly, whose try was confirmed by TMO to drag his side right back into the contest.
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Camille Lopez made no mistake with his penalty on the 25-minute mark by kicking straight down the middle, and things soon got even better for Clermont right before the interval thanks to Noa Nakaitaci's try.
Keatley successfully converted another penalty for Munster three minutes after the restart, although the Red Army were left with their work cut out having had Felix Jones sent to the sin bin following his challenge on Lopez, and they were made to pay when the fly-half kicked over for the second time.
Munster managed to hold on to deny their opponents adding to their try count moments later, but Lopez was on hand to compete his hat-trick of penalties to ensure that the home side's lead remained well intact.
The final 10 minutes saw both sides cross over the chalk; Chouly gathering Lopez's crossfield kick and racing over for the French outfit, before Duncan Casey intercepted Fritz Lee's pass to touch down for what was merely a consolation in the end.
There was still time for Keatley to land a penalty from 45 metres out, holding his nerve to see the Irish club leave Parc des Sports Marcel Michelin with a point for their efforts.
Clermont are now in command at the summit with two games remaining of the pool stage, while Munster drop down to third place.