He downed Kim Min Su of Korea in a thrilling shoot-off for the men's individual recurve bronze.
Das lost 4-6 to Japan's Furukawa in a hard-fought contest.
Deepika crumbled at the crunch as An San won 6-0, clinching the first three sets.
Atanu Das will next meet Takaharu Furukawa of Japan on Saturday for a place in the quarterfinals of the individual competition.
The Indian number one had earlier seen off a tough challenge from Chinese Taipei's Deng Yu-Cheng in a five-setter to move into round two.
Following the early exit of Atanu Das, Pravin Jadhav and Tarundeep Rai also crashed out in the round-of-32 stage of the men's individual event in Tokyo 2020 in Tokyo on Wednesday.
The Indian team of Atanu Das, Pravin Jadhav and the seasoned Tarundeep Rai had defeated Kazakhstan 6-2 in a close encounter to enter the last eight.
World No. 1 Deepika failed to notch up a single perfect 10 from eight arrows while Olympic debutant Jadhav misfired a six in the crucial fourth set after drilling in three perfect 10s.
The debutant was preferred by the Indian think-tank ahead of her husband and more experienced Atanu Das based on form shown in Friday's qualification round.
Gomboeva quickly regained consciousness after collapsing but needed to be stretchered out of the arena by medical officers.