Bess removed Cheteshwar Pujara in a timely breakthrough for England, just when his partnership with Rishab Pant was threatening to look ominous for the vistors.
Rishabh Pant's flair and Cheteshwar Pujara's doggedness came to India's rescue after a top-order collapse as the side reached 154/4 at tea in reply to England's enormous first innings total of 578 on the third day of the opening Test here on Sunday.
Pujara (53) and Pant (54), heroes of India's remarkable Brisbane Test triumph in the recent tour of Australia, combined to keep India afloat with a partnership 81 runs in 14.3 overs so far.
The home team still trails England by 424 runs.
For England, off-spinner Dom Bess struck twice in the space of two overs to give the visitors the upper hand. First, he had Virat Kohli (4) edge a delivery, which turned and bounced, for Ollie Pope to take a sharp catch at forward short-leg.
In his next over, Ajinkya Rahane (1) stepped out to hit a full toss which was dipping, leaving Joe Root to pull off a stunning one-handed catch in the short cover region.
Pant, who walked in at the fall of Rahane's wicket, played in his inimitable style and took the attack to the English bowlers.
After a cautious beginning, he went after Jack Leach and tonked four sixes and reached his half-century in 40 balls. He also smacked four boundaries.
Pujara, on the other hand, was his usual watchful self, but played some lovely shots off the spinners, skipping down the track at times.
Earlier, pacer Jofra Archer had helped consolidate England's position, removing both openers as India went to lunch at 59 for 2 in reply to the hosts' mammoth score.
The home team lost opener Rohit Sharma (6) early, edging one to Jos Buttler after Archer got the ball to move a shade along with extra bounce.
India suffered a second jolt when the promising Shubman Gill (29) was brilliantly caught by veteran James Anderson at mid-on when he failed to keep the on-drive down.
Earlier, England posted a mammoth 578 in their first innings, losing the last two wickets in the morning session after adding 23 runs to their overnight score.
Bess (34) was the first to go, trapped leg before by Jasprit Bumrah with one that angled in sharply.
After that Jack Leach (14 not out) and No.11 James Anderson (1) prolonged the England innings by five more overs before the latter was bowled when he failed to connect a slog-sweep off Ravichandran Ashwin.
Ashwin, who bowled 55.1 overs, the most by him in an innings so far, finished with 3 for 146 for his efforts while Bumrah also ended up with three wickets for 84 runs from his 36 overs.
Ishant Sharma (2/52) and Shahbaz Nadeem (2/167) took two wickets each.
Root's marathon 218-run knock was the cornerstone of England's massive total on a docile pitch after the visitors opted to bat on winning the toss.