India's number one singles player at 156, Ramkumar notched up a 6-4, 6-4 win over World No. 213 Mathieu in a 68- minute encounter.
Ramkumar slammed 16 aces and converted two break points out of three that came his way and also did not commit any unforced errors to emerge victorious.
He will next take on another Frenchman Nicolas Mahut in the second round.
The 22-year-old from Chennai is the only Indian remaining in the fray after both Yuki Bhambri and Prajnesh Gunneswaran bowed out in the opening round of qualifiers.
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