Matching 2009 champion Tsonga with nine aces, Dodig saved all three break points he faced.
Dodig will attempt to earn a place in his third semifinal of 2013 against Jarkko Nieminen of Finalnd, who beat Michal Przysiezny of Poland 6-7 (4), 6-4, 6-2.
In other matches, third-seeded Milos Raonic of Canada advanced to the second round with a 6-4, 7-6 (0) win over wild card Go Soeda of Japan in the opening match on Center Court.
Raonic, coming off a win at the Thailand Open on Sunday, dominated Soeda with his serve, hitting 24 aces.
"I served really well," Raonic said. "It was a bit earlier in the day than what I'm used to but I played well on the important points and did what I needed to do."
Raonic lost in the final last year to local favorite Kei Nishikori, who will face Feliciano Lopez of Spain in the second round on Thursday.
The fourth-seeded Nishikori is on the verge of becoming the first Japanese player to break into the top 10 and is bidding to win his second title of the season.