Murray has now won his last 19 matches on grass, stretching back to the 2012 Wimbledon final.
He will play Radek Stepanek, who beat Bernard Tomic 7-6 7-6, in round three on Thursday.
All eyes had been on practice court three earlier in the day, when Murray had a 45-minute session under the watchful eye of Mauresmo.
The Frenchwoman, 34, arrived in London on Tuesday, two days after Murray had announced they were to work together for the grass-court season initially. She took her place alongside Murray's regular team to see the world number five make an encouraging start to his campaign on grass.
On a hot afternoon, the Scot served well as he saved three break points and came through in one hour and 26 minutes.
Murray, 27, looked more sure-footed than the Frenchman from the start, recovering from 40-0 down to get the decisive break of the first set in the opening game.
There was a brief moment of alarm when the third seed took a tumble chasing down a drop shot early in the second but he brushed himself down, setting up the crucial break in game seven with a backhand down the line.
Murray saw off a late fightback from Mathieu, and two break points, before clinching victory on his third match point with another of those trademark backhand winners.
Asked about his new coach's input, Murray said: "We spoke a bit about the match and the tactics and then chatted a little bit afterwards, but this week there's not going to be any big changes in my game.
"I also wouldn't expect any before Wimbledon, but we'll definitely work on some things after the tournament is finished here.
"We will chat about the stuff that I will be working on over the next few days, and then when I get the chance to after the tournament is finished here, I'll get four or five days of practice where I can work on some things."
Top seed Stan Wawrinka benefited from an early retirement by wildcard Marcos Baghdatis, who was injured in the first set.
Former finalist Jo-Wilfried Tsonga takes on David Goffin later on Wednesday.
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