Murray had suffered a shock defeat against Mahut at the same stage of the pre-Wimbledon warm-up event 12 months ago, but the US Open champion's dominant display ensured there would be no repeat.
The 26-year-old will have little time to savour his win as he is scheduled to return to court later on Thursday to face Australia's Marinko Matosevic in the last 16 as the tournament catches up following several days of bad weather.
This was the Scot's first competitive action for a month, but he showed few signs of rust despite the slippery conditions on the grass courts in west London.
"The back felt great, especially with the conditions we have had," Murray told the BBC.
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"It is very cold and I'm sure, as anyone who has had a back problem knows, that's not good. But even with all the stopping and starting it feels great and that is probably the most satisfying thing about the match."
But top seed Murray had taken the first set on Wednesday and was level at 2-2 in the second before the third rain interruption of the day caused the match to be postponed.
When they finally resumed 24 hours later, Murray found himself under pressure and he had to save three break points in the early exchanges.