It was England's first Test win in India since 1985.
India were shot out for a pathetic 100 off just 48.2 overs in their second innings while chasing a stiff 313-run target on the fifth and last day.
They lost their last seven wickets for 25 in a dramatic afternoon collapse here on Wednesday.
England had lost the second Test by nine wickets at Mohali after drawing the first at Nagpur, but raised their game when it mattered most. Their final-day heroes were Flintoff and Udal.
Fast bowler Flintoff finished with 3-14 off 11 tight overs in his first Test series as captain and was brilliantly supported by off-spinner Udal (4-14), playing only his fourth Test at the age of 37.
The tourists were without key players Michael Vaughan, Marcus Trescothick, Simon Jones and Ashley Giles, and pace spearhead Steve Harmison was also absent for this Test. But they were superbly led by Flintoff who used his bowling resources ably in hot and humid conditions.
He was named both man of the match and man of the series.
England virtually had the match in their pocket when they removed key batsmen Rahul Dravid and Sachin Tendulkar in the opening two overs after lunch.
The in-form Dravid stonewalled for 60 balls to score just nine runs before being caught behind off Flintoff while Tendulkar was smartly caught by Ian Bell at short leg off Udal.
Tendulkar, booed by his home crowd after his first-innings dismissal for one run, contributed only 34. He has been without a half-century since his world record 35th Test hundred against Sri Lanka at Delhi last December.
India, chasing a record total to win a Test at home, were then struggling to save the game under pressure but the task always looked beyond Virender Sehwag, Yuvraj Singh and Mahendra Dhoni.
Sehwag walked in with a runner (Wasim Jaffer) because of back spasms, but was soon trapped leg-before for a duck. He batted at number seven instead of opening the innings because he was off the field on Tuesday.
Dhoni lived dangerously, playing one shot too many against a disciplined England attack. He was dropped on three by Monty Panesar at mid-on off Udal but fell to the same combination in the same over.
England then made short work of India's tail, with Udal sparking celebrations in England's camp with the wicket of last-man Munaf Patel, caught by Matthew Hoggard in the deep.
There was no hint of the coming collapse in the morning session in which India had lost two wickets for 57 runs after resuming on 18-1.
Nightwatchman Anil Kumble went in the day's third over, trapped leg-before by a Hoggard delivery which appeared to be missing the leg-stump. He failed to add to his overnight score of eight.
Jaffer (10) scored his first run after 23 balls when he square-drove James Anderson for a four. But it turned out to be his only display of aggression before he fell soon afterwards, trapped leg-before by Flintoff.