The Frenchman ground out the win in a tight affair, coming through in two tie-breaks 7-6 (7-2) 7-6 (7-3).
Neither player dropped serve as they each proved adept at surviving break-point chances, Mathieu seeing off eight while German Becker held his nerve on seven.
But it was Mathieu that had the greater quality in the two breakers as he progressed through to the next stage of the ATP 250 tournament, which has had a French winner in each of the past five years.
Mathieu - runner-up in 2008 - faces a much tougher test in the second round, where he will meet big-serving Czech Lukas Rosol.
In Monday's only other first-round clash Edouard Roger-Vasselin failed to make it a perfect day for French supporters on the hard courts in Metz.
Roger-Vasselin - a winner in the doubles with Nicolas Mahut in 2012 - came undone at the hands of Luxembourg's Gilles Muller, who emerged victorious 6-1 7-6 (8-6).
It was Muller's serve that made the difference as he knocked down an impressive 18 aces to set up a clash with last year's runner-up and top seed Jo-Wilfried Tsonga.