The South Korean won the first game 21-18 in a close fight in Hangzhou that sent both players diving to the ground to return shots.
Top-ranked An hurt her knee during the first game and was attended to by medics.
The scrape seemed to break her momentum in the second game, which Olympic champion Chen won 21-17.
The South Korean staged a remarkable comeback in the third, crushing her opponent 21-8 despite most of the audience cheering for the Chinese player.
An emotional An embraced her coaches after her victory, with both players limping off court.
In an incredible 2023, the 21-year-old An has risen to number one for the first time and now won 10 titles, including the world championship.
"No matter what, I worked hard to play this match well and I didn't overthink it," An told AFP.
Silver medallist Chen said the 90-minute match had left her with blisters on her feet and cramp in her legs.
"I'm still very happy because the silver medal is still something I worked hard to earn," she said, adding that she had "a lot to learn" from An.
The home crowd still had plenty to celebrate as Li Shifeng beat fellow Chinese player Shi Yuqi to win the men's singles title.
Li defeated former world silver-medallist Shi 23-21, 21-13 and celebrated by ripping off his shirt and performing a backflip.
In the men's doubles final, Chirag Shetty and Satwiksairaj Rankireddy made history for India with their country's first badminton gold at the Asian Games ever.
China won the mixed doubles and women's doubles.
