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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.
The gruelling schedule will be a significant test for two-time Queen's champion Murray, who missed the recent French Open after aggravating a recurring lower back injury during a match against Marcel Granollers in Rome.
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.
"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."
Mahut, a former Queen's finalist who played the longest match in tennis history against John Isner at Wimbledon in 2010, can be a tricky proposition on grass.
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.
But, in chilly conditions, he turned up the heat on Mahut in the tie-break, producing two sublime passing shots to finish off the match after 43 minutes.
Lleyton Hewitt continued his giant-killing run as the Australian moved into the quarter-finals with a 5-7, 6-3, 6-4 victory over American sixth seed Sam Querrey.
Hewitt is a four-time champion at the pre-Wimbledon warm-up event, but thoughts of a record fifth crown were some way from the former world number one's thoughts when he arrived at the start of the week.
The 32-year-old is well into the twilight of his career and his lowly 82nd position in the world rankings reflects that diminished status.
But the former Wimbledon champion has always thrived on grass and he produced one of his best performances of 2013 to brush aside the highly-regarded Grigor Dimitrov in the second round before cutting big-serving Querrey down to size on Thursday.
Hewitt will face former US Open champion Juan Martin del Potro or British wildcard Daniel Evans in the last eight.
Czech second seed Tomas Berdych maintained his strong start to the tournament as he progressed to the quarter-finals with a 6-3, 6-4 win over Slovenian 16th seed Grega Zemlja.
World number six Berdych, a Wimbledon finalist in 2010, next faces defending champion Marin Cilic or Feliciano Lopez.
Jo-Wilfried Tsonga, the French fourth seed, finally won his delayed second round tie against compatriot Eduoard Roger-Vasselin.
Tsonga, fresh from his first semi-final appearance at the French Open, was pushed hard before eventually winning 6-3, 6-7 (2/7), 6-3 in a tie postponed on Wednesday due to rain.
The world number seven, who plays Holland's Igor Sijsling in the last 16, admitted he had found it hard to adjust to the slippery grass court and unseasonably cold weather.
"The conditions are not easy. It is freezing but I tried to get used to the grass and play my game," he said.
Results from the fourth day of the ATP tournament at Queen's Club (x denotes seeding):
Second round:
Igor Sijsling (NED) bt Denis Istomin (UZB x14) 7-6 (7/4), 6-1
Feliciano Lopez (ESP) bt Julien Benneteau (FRA x11) 7-6 (7/5), 6-4
Jo-Wilfried Tsonga (FRA x4) bt Edouard Roger-Vasselin (FRA) 6-3, 6-7 (2/7), 6-3
Andy Murray (GBR x1) bt Nicolas Mahut (FRA) 6-3, 7-6 (7/4)
Third round:
Tomas Berdych (CZE x2) bt Grega Zemlja (SLO) 6-3, 6-4
Lleyton Hewitt (AUS) bt Sam Querrey (USA x6) 5-7, 6-3, 6-4
Juan Martin del Potro (ARG x3) bt Daniel Evans (GBR) 6-0, 6-3
Denis Kudla (USA) bt Kenny De Schepper (FRA) 6-1, 6-3
Marin Cilic (CRO x5) bt Feliciano Lopez (ESP) 4-6, 7-6 (7/5), 7-5
