The third-seeded Isner, the highest-ranked American (10th) in the world rankings, was pushed to three sets before beating Slovakia's Martin Klizan 4-6, 6-3, 7-5.
Tsonga, the highest-ranked player (sixth) in the final hard-court tournament before the U.S. Open, beat Brazil's Thomaz Bellucci 6-3, 7-6 (7-2).
Isner, who was born in nearby Greensboro, N.C., showed the effects of a two-week layoff, coming out sluggish against Klizan.
The left-hander broke Isner's serve in the opening game of the first set, and kept him scrambling with his groundstrokes.
However, Isner eventually picked up his game, serving 13 aces -- the fastest recorded at 140 mph -- and breaking Klizan's serve late in each of the final two sets.
"I felt the match was going to be tough, and that's exactly what it was," Isner said. "He was playing pretty well, and I wasn't on top of my game. That's all it took in that first set, and from that point on I knew it was going to be a battle.
"I wasn't that great out there today, but as the tournament progresses, I should get stronger."
Tsonga, a semifinalist at Wimbledon earlier this season, broke Bellucci's serve in the first set. But the Brazilian overcame four double faults to force a second-set tiebreaker.
"It was a good match for me," said Tsonga, who teamed with Michael Llodra for the silver medal in men's doubles at the London Olympics. "The second set, I had many occasions to finish it, but I did much better in the tiebreaker."
In addition to Andujar, three other seeded players were eliminated Tuesday: eighth-seeded Julien Benneteau of France, last year's Winston-Salem finalist; 11th-seeded Denis Istomin of Uzbekistan and 12th seed Kevin Anderson of South Africa.
Benneteau lost to Poland's Lukasz Kubot 2-6, 7-5, 6-4, his second straight win over a seeded player; Istomin lost to Belgium's Steve Darcis 6-2, 6-2; while Anderson fell to qualifier Ernests Gulbis of Latvia 6-4, 7-6 (7-2).
Also advancing to Wednesday's round of 16 were second-seeded Tomas Berdych of the Czech Republic, fourth-seeded Alexandr Dolgopolov of the Ukraine, sixth seed Marcel Granollers of Spain, seventh-seeded Sam Querrey, ninth-seeded Feliciano Lopez of Spain, 13th seed Jurgen Melzer of Austria, 14th seed David Nalbandian of Argentina and 16th seed Jarkko Nieminen of Finland.
Berdych beat Alex Bogomolov Jr. of Russia 6-3, 7-5; Dolgopolov downed Taiwan's Yen-Hsun Lu 6-1, 6-3; Granollers won 6-4, 6-2 over Ryan Harrison; Querrey defeated Colombia's Santiago Giraldo 6-3, 6-2; Lopez defeated Donald Young 6-2, 6-3; Nalbandian beat Robin Haase of the Netherlands 6-2, 6-4; and Nieminen was a 6-3, 6-3 winner over Germany's Benjamin Becker.