The left-handed Duckett reached his third Test hundred in 88 balls as England reached 207-2 to cut their deficit after they bowled out India for 445 on day two in Rajkot.
Duckett, on 133, and Joe Root, on nine, were batting at close of play. The tourists still trail India by 238 runs.
Duckett came out roaring and, despite a hostile opening spell by the Indian quicks, hit regular boundaries in his 80-ball opening stand of 84 with Zak Crawley.
Ashwin took down Crawley after tea for his 500th Test wicket, becoming only the ninth international bowler and second Indian after Anil Kumble (619) to reach the milestone.
"It's a good time to celebrate, probably go have dinner and think about how far I have come in life," Ashwin said.
Crawley top-edged a delivery going down the leg side to be caught at fine leg as Indian players hugged Ashwin, who waved to the applauding crowd.
But Duckett kept up the charge and, with Ollie Pope at the other end, raced to his ton with a boundary off Mohammed Siraj, who sent back Pope lbw to end a 93-run stand.
"To be that far behind in the game and go out and play like that showed real bravery and skill," England quick Mark Wood said of Duckett's knock.
"The way India changed the field and then he'd hit it somewhere else, it was such a skilful innings against a good attack."
"I'm delighted, the way him and Zak go out together and put pressure on the opposition," said Wood, who took four wickets in India's innings.
'Poor motor skills'
The tourists began at five for no loss after India were penalised five runs for Ashwin running on the pitch, earning a reprimand from on-field umpire Joel Wilson.
"My poor motor skills did not allow me to get off the pitch in time," Ashwin said. "If the English media and players think it was on purpose, it wasn't."
Ashwin (37) and debutant Dhruv Jurel (46) put on a stubborn eighth-wicket stand of 77 before England wrapped up the Indian innings in the second session.
Both men batted with patience and occasional boundaries to tackle a persistent England attack which was rotated by skipper Ben Stokes in his 100th Test.
Rehan Ahmed sent back Ashwin with his leg spin in the afternoon and then got Jurel caught behind on a delivery that pitched and turned sharply.
Number 10 Bumrah hit a breezy 26 to frustrate England before fast bowler Mark Wood got him lbw to finish with 4-114 after his three strikes on day one.
Fast bowler James Anderson struck first to get nightwatchman Kuldeep Yadav caught behind for four. Anderson moved to 696 Test wickets over 185 matches since his 2003 debut.
Root had overnight centurion Ravindra Jadeja caught and bowled on 112 as the hosts lost two wickets in the first five overs of the day.
The five-Test series is level at 1-1 after England won the opener in Hyderabad and India bounced back in the second match.
