Stand-in India captain Jasprit Bumrah did the bulk of the damage by hitting 29 runs off the bat in an over from Broad that also featured five wides and a no-ball struck for six.
An on-field altercation with India's Virat Kohli appeared to fire up Bairstow who registered his fourth consecutive Test century. No other England batsman made more than Sam Billings' 36, with the wicketkeeper one of four wickets for Siraj.
England, set 378 to win, finished on 378-3 with Joe Root 142 not out and Jonny Bairstow 114 not out.
India had already made sure of a series success after taking an unassailable 2-0 lead in a three-match campaign with victories at Southampton and Edgbaston.
All-rounder Hardik Pandya (4/24) and leg-spinner Yuzvendra Chahal's (3/60) superb bowling effort helped India bowl out England for 259 in the series-deciding third and final ODI at Old Trafford.
Reece Topley took 6-24, the best figures by any England bowler in a One-Day International (ODI), as the world champions bounced back to beat India by 100 runs at Lord's on Thursday.
Fans had gathered at cafes and markets across India to watch the semi-final, hoping the team would win to set up a blockbuster final against Pakistan.
A resolute opener Rohit Sharma smashed a gritty hundred to lead India’s fight-back against England in the fourth Test on Saturday. India were 270-3 at stumps in their second innings when bad light forced an early stop of the play on the third day of the fourth Test.
The BCCI said Shastri, bowling coach B Arun, fielding coach R Sridhar and physiotherapist Nitin Patel were all isolating as a "precautionary measure" following 59-year-old former India all-rounder Shastri's positive lateral flow test late on Saturday.