Gasquet opened the best-of-five series with a 7-6 (7-3), 6-2, 6-2 victory over Nick Kyrgios and Tsonga defeated Lleyton Hewitt 6-3, 6-2, 7-6 (7-2).
Gasquet won an impressive 80 percent of the points when serving and faced just one break point. He also committed only 27 unforced errors, compared with 52 by Kyrgios.
Neither player had a break opportunity in the first set but Gasquet won 2-of-4 of his chances in the second set and 2-of-3 in the third.
Tsonga also went through his match without dropping serve, winning 75 percent of the points on serve. He moved ahead with one break in the first set and had a pair in the second.
He broke open the tight third set with a string of points in the tiebreaker for the win.
If Hewitt and Guccione win, Hewitt is scheduled to return to the court against Gasquet for the first singles match Sunday followed by a Tsonga-Kyrgios pairing to decide the series.