The Briton, 29, saved set points in both sets as he won 7-6 (10-8) 7-6 (7-1) to reach round two of the Aegon Championships.
Lendl was courtside for the first time since 2014, having agreed last week to resume their working relationship.
Murray goes on to face fellow Briton Aljaz Bedene or Frenchman Benoit Paire.
Asked about Lendl, Murray told BBC Sport: "He doesn't really give many signs during the match. He's got quite a straight face.
"I started working with Jamie Delgado a few months ago and that's worked extremely well, and obviously I had some of my best years with Ivan a few years ago.
"I think it's a very strong team and I'm hoping they can help me win more major events."
Swiss second seed Stan Wawrinka earlier lost 6-2 7-6 (7-3) to Spain's Fernando Verdasco.
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Lendl had resumed his coaching role on Tuesday morning, directing the drills as Murray went through their first practice together for over two years.
The session was cut to just an hour because of the rain which has blighted the first two days at Queen's Club, leaving Murray a little short of time on the grass.
Mahut, ranked 51st, won a title on the surface 24 hours earlier in the Netherlands and looked the more sure-footed when the pair finally got on court just after 17:00 BST.
The Frenchman broke serve first but could not convert the next game from 40-0, and Murray came through a tense tie-break on his third set point.
Mahut had an opportunity at 8-7, failing to make a return of serve, but he moved 4-1 up in the second set and a furious Murray went close to smashing his racquet.
The Scot complained to the umpire about the conditions underfoot, and Mahut would later slip over, but Murray fought back to 5-5 and saved three more set points before dominating the second tie-break.
"It was tough," said Murray, a four-time winner at Queen's.
"Nico's a very accomplished grass-court player and because of the weather the last few days, we haven't been able to get on the grass too much to hit.
"It was a tricky first round and I'm glad I managed to get through."
Wawrinka, 31, never looked comfortable on a greasy surface and struggled to cover the powerful forehand of his opponent.
Verdasco wrapped up the match - which was interrupted by rain - as Wawrinka's serve faltered during a tie-break.
"For sure conditions are difficult for everybody since a few days with the rain," said world number five Wawrinka.
"That's the same for everybody, but it was a tough match.
"Grass is always a challenging surface for my game. I need quite a lot of practice to play my best game."

Substances like Mamba and Spice can have serious side effects such as paranoia, psychosis and seizures, and have been linked to UK deaths.
Previously known as legal highs, many of the drugs are untested so the risks of taking them is unknown.
There have been a number of cases in towns such as Wrexham, with a lawyer saying there is a "big problem" there.
Crimestoppers is now offering a reward for information on the supply of these drugs.
It wants to hear from people who know of individuals who are manufacturing, importing or supplying these drugs to people in the area, including the homeless.
"These substances are relatively new on the streets, so information is hard to come by as to who is producing and supplying them," said Gary Murray, the charity's north Wales regional manager.
"We understand there is a stigma attached to 'grassing' on people, but these substances are dangerous, to the individuals taking them and to the community around them."

The proposed walking and cycling path between Carmarthen and Llandeilo is expected to cost between £5m and £8m.
On Thursday, Carmarthenshire's planning committee approved the western phase between White Mill and Nantgaredig.
It is estimated the path could attract over 15,000 visitors a year, generating up to £2m in the local economy.
Work on the first section, between Fronun and Bwlch Bach, started in December 2016.
The council said negotiations with landowners were ongoing with the scheme being developed in phases.
Andy Murray gave coach Ivan Lendl the perfect welcome back with a fine win over France's Nicolas Mahut at Queen's Club in London.
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A charity has launched an appeal to help stop the supply of "dangerous drugs" in north Wales.
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The second section of the 16-mile (25.7km) Tywi Valley Path in Carmarthenshire has been given the go-ahead.