Stokes made a remarkable 155 before he was dismissed with England 70 runs shy of a steep target of 371.
England were eventually dismissed for 327 as holders Australia, bidding for a first away Ashes series win in 22 years, went 2-0 up in the five-match campaign.
But it was the dismissal of Bairstow shortly before lunch on the last day that lit the blue touch paper on this match and incited the fury of the usually sedate Lord's crowd.
Bairstow, the last of England's specialist batsmen, fell in bizarre fashion for 10 when he wandered out of his ground after ducking a Cameron Green bouncer and quick-thinking wicketkeeper Alex Carey under-armed the ball onto the stumps.
Bairstow thought he had secured his ground by tapping his bat behind the crease.
Australia captain Pat Cummins could have withdrawn the appeal but the decision was referred to third umpire Marais Erasmus, who ruled Bairstow had been stumped, with England now 193-6.
A usually sedate Lord's crowd reacted in fury with a chant of "Same old Aussies, always cheating" in a reference to Australia's 2018 ball-tampering scandal in South Africa.
Incoming batsman Stuart Broad told Carey "you'll be forever remembered for that".
But a spokesman for Marylebone Cricket Club (MCC) the owners of Lord's and the guardians of cricket' Laws, later told AFP Bairstow had been given out correctly.
An Australia team spokesman later alleged players had been "verbally abused" and "physically contacted" by irate MCC members in the Lord's Pavilion during the lunch break.
An angry Stokes, then on 62, hit out and on 77 he slammed a fierce pullback at Cummins which the fast bowler dropped.
Stokes then hooked Green for three sixes off successive deliveries - the second dropped over the boundary by Mitchell Starc - to complete a stunning 142-ball hundred.
Stokes accepted Bairstow had been out but he also told the BBC: "Would I want to win a game in that manner? The answer for me is no."
Stokes hit nine fours and nine sixes in all but said his aggression had not been motivated by anger at Bairstow's exit but by his "best option" being to take the fielders out of play.
'Tried it on'
Cummins insisted there was "nothing untoward or sneaky," about Bairstow's dismissal.
"Jonny was leaving his crease every ball," he said. "He did it four or five balls. You're meant to stay in your crease in cricket. Bairstow has tried it with a lot of our guys."
At lunch, England were 243-6, with Stokes a remarkable 128 not out.
Stokes continued his spectacular assault two balls after the interval by lofting Josh Hazlewood for a brilliant straight six.
Two balls later, however, he was dropped at deep backward square when Smith grassed a routine chance before he was again missed on 114 by a diving Carey.
Left-handed batsman Stokes struck two more sixes off Hazlewood, the second a remarkable one-handed hit.
Australia were without Nathan Lyon after the off-spinner, who had limped out to bat at No 11 on Saturday, was off the field with the severe calf tear he had suffered while fielding Thursday.
Stokes, however, was unable to repeat his heroics of 2019 when the all-rounder's astounding unbeaten century secured a thrilling one-wicket win in an Ashes Test at Headingley, with England reaching a total of 359 - their highest fourth-innings winning chase against Australia.
He fell when he skyed Hazlewood to Carey to end a 214-ball innings including nine fours and nine sixes. Stokes walked off to a standing ovation but at 301-7 the game was all but up for England.
Ollie Robinson and Broad, who made 11 in a stand of 108 with Stokes, fell in quick succession before Starc ended the match by bowling Josh Tongue
England now have a mountain to climb heading into Thursday's third Test at Headingley.
Only once have a team come from 2-0 down to win a Test series, when an Australia side inspired by batting great Don Bradman, recovered to take the 1936/37 Ashes 3-2.
