Gasquet won 7-6 (7-3) 7-6 (7-5) in just over two hours to reach the quarter-finals of the ATP 250 event in Bogota, where he will face eighth seed Victor Estrella Burgos of the Dominican Republic.
While Groth sent down 20 aces and won 84 per cent of points on his first serve, Gasquet had some joy on the Australia's second serve and claimed crucial breaks in both tie-breakers to win his first match of the tournament.
Gasquet won 41 per cent of points on Groth's second serve.
The 28-year-old Frenchman had a bye through the first round at the Claro Open but started strongly on Thursday, winning the opening game to love on his serve.
It would be a familiar refrain throughout the match, however, with eight of the 24 games finishing to love, as neither player was able to convert a break point.
In the first-set tie-breaker, Gasquet took a 2-1 lead when he won Groth's second service point and then won the last four points to claim the set.
The second-set tie-breaker was closer with both players claiming two-point leads early before Groth levelled at 5-5, only for Gasquet to win the next two points - the last on the Australian's serve - to triumph in 2:06.
Next up for Gasquet will be world number 92 Estrella Burgos, who cruised past Canada's Peter Polansky 6-4 6-1.
Inaugural Claro Open champion Ivo Karlovic of Croatia remains in the hunt for back-to-back titles after accounting for Israel's Dudi Sela 7-6 (10-8) 7-6 (7-5), while he will take on Chinese Taipei's Jimmy Wang next, who defeated the United Kingdom's James Ward 6-3 6-1.