The score earned France a bonus point at a sodden McLean Park in Napier.
Coach Marc Lievremont may have made 11 changes following the unimpressive victory over Japan, but this was another disjointed performance from Les Bleus, although their cause was not helped by difficult conditions following a torrential downpour prior to kick-off.
Scrum-half Morgan Parra converted tries from Clerc (three) and Damien Traille, as well as booting five penalties for a 23-point haul with Francois Trinh-Duc adding a drop goal.
Canada, who had beaten Tonga in their opening game, claimed a try through centre Ryan Smith with fly-half Ander Monro dropping two goals. Full-back James Pritchard kicked two penalties and converted Smith's score but was off target with three other efforts at goal.
Bedford full-back Pritchard had fired the Canucks into a third-minute lead after France riskily tried to play their way out from deep in the heavy rain.
But Pritchard was at fault for the opening try moments later as his failure to gather a Trinh-Duc up and under gave Clerc an easy run in, Parra adding the simple conversion.
Canada hit back in the ninth minute with a similar score. Monro put up the bomb and Traille failed to deal with it in the shadow of his own posts, the resulting scramble allowing Smith to dot down for a converted score.
A missed penalty and drop goal from Pritchard and Monro respectively saw them fail to extend their lead but an accurate tactical kicking game was turning up plenty of French errors.
Parra drew France level with a penalty of his own after 15 minutes before Pritchard missed another valuable three-point chance for the underdogs.
Les Bleus' discomfort under the high ball was again highlighted as Louis Picamoles surrendered possession, but France were able to end a scrappy half in front thanks to three Parra penalties late in the piece.
Pritchard's poor night in front of the posts continued as he missed another penalty shortly after the restart, although Monro eased the blow by landing a drop goal to trim the lead to six points.
Parra quickly responded with his fifth penalty but was off target for the first time with his sixth effort following Monro's second successful drop at goal.
The tit-for-tat scoring continued as a Trinh-Duc drop goal was cancelled out by a Pritchard penalty, with France unable to shake off the battling north Americans.
But they were finally able to put daylight between the sides as Traille cut a great line from full-back before dummying and cutting inside to score by the posts for Parra to convert.
And the fading Canadians then ran out of defenders in the final minutes to allow Clerc to slide over on the right, before the Toulouse wing completed his treble in injury time following a superb break from Maxime Mermoz to tee France up for their clash with hosts New Zealand next weekend.
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