Anderson removed captain Rahul Dravid and Harbhajan Singh and then brought off a run-out on Monday to help reduce India to 217-8 at tea on the third day in reply to England's first-innings total of 400.
Anil Kumble was unbeaten on nine at the break with Shanthakumaran Sreesanth, who had yet to open his account.
England bowled and fielded brilliantly in a match they need to win to square the series. They drew the first Test at Nagpur before losing the second by nine wickets at Mohali.
Wicket-keeper Geraint Jones backed the bowlers brilliantly, taking three superb catches on Monday to add to his two on Sunday.
Dravid (52) and wicket-keeper Mahendra Dhoni (64) offered some resistance to the England attack but could not lift their team out of trouble.
England began impressively when fast bowlers Andrew Flintoff and Anderson each grabbed a wicket in the morning session to account for overnight batsmen Yuvraj Singh (37) and Dravid.
Flintoff tasted success in his opening over when he removed Yuvraj after the hosts had added just five runs to their overnight total of 89-3.
Left-handed Yuvraj was out to a poor shot as he attempted to cut a ball going outside the off-stump, only to see Jones diving in front of first slip to bring off a low catch.
India suffered a major setback when they lost the well-set Dravid, caught down the leg-side by a diving Jones off Anderson. He fell shortly after completing his 42nd Test half-century.
Dhoni made some aggressive shots during his second Test half-century before being given run out by TV umpire K. Hariharan of India, following Anderson's direct hit to the non-striker's end from mid-on.
Anderson claimed two of the five Indian wickets that fell on Monday but he deserved more.
was the unlucky bowler when Monty Panesar dropped Dhoni at mid-off and then substitute Matt Prior let off Dravid at gully in the space of three deliveries. Dhoni was then on 23 and Dravid on 52.
Anderson capped his team's impressive show in the first two sessions with the wicket of Harbhajan, who fell to an overhead catch by Jones just before the tea break.
The Indian batsmen had had a torrid time in the morning session, with swing bowler Matthew Hoggard and Flintoff posing problems with pace and bounce in their opening spells.
Dhoni got a nasty blow on the helmet while ducking into a Flintoff bouncer, but recovered quickly to pull the next ball for four. He continued playing shots under pressure, hammering Flintoff for another four in the next over.