The world number eight only returned to action last week having picked up a knee injury at Wimbledon in June.
And Tsonga, who beat Gael Monfils in the first round, struggled throughout as a dogged Dodig forced the Frenchman into a number of unforced errors.
Dodig took the second set and match courtesy of a Tsonga forehand that found the net in the tie-break.
"I'm not shocked by this result," the second seed told reporters.
"This game is difficult enough with two legs. With one and a half it's complicated."
Top seed Juan Martin del Potro overcame Carlos Berlocq 6-2 6-2 to secure his spot in the quarter-finals.
Del Potro just lost eight points on his serve to and converted four of his seven break point opportunities to ease into the next round.
"I played better than yesterday, which was pleasing," said del Potro. "I hit my forehand really strong and I was confident, able to break serve many times.
"Playing under a closed roof is a little different to the outdoor conditions, but it helps my game and especially my serve."