Terry looked set to enjoy some respite from the Football Association and police probes into allegations he racially abused QPR's Anton Ferdinand when he gave the Blues a 2-1 half-time lead at Stamford Bridge.
But the Gunners staged a sensational second-half comeback to turn the game on its head and, though Juan Mata equalised at 3-3, Terry's mistake allowed Van Persie to make it 4-3 before the Dutchman completed his treble in stoppage time.
It handed the visitors their first Premier League away win since April and left Chelsea six points behind leaders Manchester City having played a game more.
That piled on the misery for the Blues, who have been under siege in the past week, which saw them hit by a humiliating defeat at QPR, three more red cards, the Terry racism storm, two - potentially three - Football Association fines and a supporter revolt over their stadium plans.
Fan reaction to Terry - who categorically denies racially abusing Ferdinand - was predictable on what his first public appearance since the allegations emerged.
There were cheers from the Chelsea faithful - whose posters included messages like 'JT is not racist' and 'Captain, Leader, Legend' - and jeers and abuse from the Arsenal contingent, who ridiculed him over his current travails.
But both sets of supporters were soon transfixed by a feast of attacking football.
After Ashley Cole twice got in behind the visitors' defence, Chelsea really should have scored when three of their players beat the Gunners' offside trap but Daniel Sturridge somehow sent his cross straight at Wojciech Szczesny.
The home side also looked vulnerable and Gervinho somehow scuffed a tap-in wide after Theo Walcott surged past Cole. The Arsenal winger also picked out captain Van Persie, who volleyed over.
It was the Arsenal defence which buckled first after 14 minutes, the outstanding Mata teasing Van Persie and swinging in a right-wing cross which the unmarked Frank Lampard stooped to head home.
Van Persie was robbed after again getting in behind before the goalmouth action finally relented until the 29th minute, when Lampard's flighted ball put Sturridge clean through but the forward got his left-footed finish all wrong.
And Arsenal made him pay seven minutes later when they equalised with the kind of goal they are all too often accused of trying to score.
Ramsey picked out Gervinho's brilliant run and the Ivory Coast striker drew Petr Cech and squared for Van Persie to slot into an empty net.
Sturridge thought he had put Chelsea back in front four minutes before the break but he was rightly ruled offside poking home Ramires' cross.
But they were gifted the lead on the stroke of half-time through Terry, of all people, who got in front of Per Mertesacker to cushion Lampard's corner in off the post right in front of the Arsenal fans.
Every one of Terry's outfield team-mates joined in the celebrations with their skipper, who appeared to swear at the away support in response to their goading.
Incredibly, the tables turned within 10 minutes of the restart as Arsenal scored twice to take the lead and Szczesny avoided what looked a certain red card.
Van Persie and Ramsey both missed chances to level inside a minute but Andre Santos did three minutes later when his superb first touch onto Gervinho's ball gave him time to send a strike through Cech's legs and into the net.
Chelsea immediately went on the attack and Szczesny appeared lucky not to be sent off when he came rushing out of goal and tripped Cole on the edge of the box.
Referee Andre Marriner showed only a yellow card and Szczesny rubbed salt into the wound by tipping Lampard's resulting free-kick wide before Arsenal scored again, Walcott bundling his way through the home defence and lashing the ball past Cech at the near post.
Both sides made a raft of changes as the end-to-end action continued, with Ramsey missing another good chance for the visitors who were looking the more dangerous.
Terry was forced to act as peacemaker after a clash between Van Persie and Branislav Ivanovic, both of whom were booked.
And there was a grandstand finish thanks to a wonderful strike from Mata, who let fly from 25 yards to beat the despairing Szczesny.
But like the first half, the second ended up being decided by Terry, who slipped chasing substitute Florent Malouda's backpass, leaving Van Persie to race clear, round Cech and make it 4-3.
Mata twice saw efforts blocked as Chelsea tried to level before Van Persie hit them on the break again by blasting left-footed past Cech.