Mannarino secured a 7-6 (9-7) 6-4 win on Monday (AEST) to reach the second round in his homeland.
The Frenchman roared back from 5-1 down in the first set and rocked Karatsev further by winning the tie-break.
Mannarino then claimed the only break of the second set to advance at the expense of the Russian.
Holger Rune will be his opponent in round two following the Dane's 7-6 (7-4) 7-6 (7-3) defeat of Arthur Rinderknech.
Francisco Cerundolo will do battle with top seed Cameron Norrie in the second round after the Argentine saw off James Duckworth 6-2 3-6 6-3.
There were just two matches in the first round of the Geneva Open, with Tallon Griekspoor and Kamil Majchrzak progressing at the expense of seeds Tommy Paul (6) and Alexander Bublik (8) respectively.