Ramiro Funes Mori's late goal appeared to have earned the Toffees a first league win at Stamford Bridge in 21 years, but Terry flicked in on his 700th club appearance.
Terry's own goal and a Kevin Mirallas strike put Everton 2-0 up, but Chelsea scored twice in quick succession to restore parity.
A howler from Tim Howard allowed Diego Costa to net and Cesc Fabregas' deflected strike levelled matters.
Worse was to follow for Chelsea when Costa limped off and Funes Mori turned in fellow substitute Gerard Deulofeu's cross.
But Terry converted after a late bombardment, to the fury of the visitors after seven minutes' stoppage time had been signalled by the fourth official, to earn a scarcely believable point.
Chelsea are now unbeaten in seven since Jose Mourinho's departure, with Guus Hiddink avoiding a first loss of his second spell in caretaker charge.
Chelsea owner Roman Abramovich was again watching on, wearing a jacket adorned with the Champions League logo, and must be wishing for this season of strife to end.
The point did little to aid Chelsea's hopes of returning to a place among Europe's elite.
That prospect appears to depend on winning the Champions League this season, a scenario which appears even more far-fetched than when they did so in 2012.
A more realistic prospect, as Hiddink admitted on Friday, is relegation and Chelsea flirted with falling closer to the drop zone.
The first half was tepid, with little action on goal.
Howard denied Willian and Bryan Oviedo shot just wide for Everton, for whom former Chelsea striker Romelu Lukaku threatened.
Lukaku, who asked Mourinho to leave and was granted his wish, lurked menacingly, but Kurt Zouma repeatedly kept him quiet, often by covering for captain Terry.
Two Everton goals in eight second-half minutes brought the contest to life and watching England boss Roy Hodgson would have been pleased with the impact of Leighton Baines.
First, the left-back crossed for Terry to bundle the ball into his own net and then, after Ross Barkley had hit the post, Baines centred for Mirallas to control and fire in on the turn to leave the Toffees looking on course to claim a first league win at Stamford Bridge since November 1994
It was all Everton, until calamity struck for the visitors.
Howard found himself in no-man's land in a mix-up with Phil Jagielka, with neither reaching Fabregas' long ball, and Costa walked the ball in.
Suddenly Chelsea had awoken from their slumber
Oscar shot at Howard and then Fabregas benefited from a deflection off Muhamed Besic to equalise.
Ten minutes after going 2-0 down, Chelsea were level.
They might have gone ahead had Costa made meaningful contact with a Cesar Azpilicueta cross.
Everton almost had a third themselves as Barkley threaded a ball through for Mirallas, who was thwarted by Thibaut Courtois.
Costa had to withdraw, with Loic Remy coming on, after the Spain international came off worse when fouling Baines.
Everton thought they had won it when Remy headed Deulofeu's corner back to him and the Everton winger crossed again to the back post where Funes Mori adjusted to flick the ball past Courtois
Lukaku was also unmarked and might have scored too
But Terry denied the Toffees a memorable win.
TWEET OF THE MATCH
"Sometimes you need luck
1 minute extra of the extra time plus it was offside" - former Chelsea manager Ruud Gullit
Thibaut Courtois: 7
Branislav Ivanovic: 6
Cesar Azpilicueta: 6
Kurt Zouma: 7
John Terry: 6
John Obi Mikel: 6
Nemanja Matic: 6
Cesc Fabregas: 8
Diego Costa: 8
Loic Remy: 6
Tim Howard: 7
Bryan Oviedo: 7
Leighton Baines: 7
John Stones: 7
Phil Jagielka: 5
Gareth Barry: 7
Aaron Lennon: 7
Muhamed Besic: 6
Romelu Lukaku: 7
Ross Barkley: 6
Kevin Mirallas: 8
Ramiro Funes Mori: 6
Gerard Deulofeu: 6
Steven Pienaar: 6
Cesc Fabregas - Made one and scored one in Chelsea's fightback as signs of his once apparently telepathic understanding with Diego Costa returned.
MOMENT OF THE MATCH
The clock was ticking onto 98 minutes when Terry flicked in an equaliser for Chelsea on his 700th club appearance
Everton protested as fourth official Craig Pawson had signalled only seven minutes would be added.
VIEW FROM THE BENCH
Chelsea made a solitary change from the fractious draw with West Brom, with Oscar replaced by Nemanja Matic, while Everton made three changes following the draw at Manchester City
Bryan Oviedo, Kevin Mirallas and Aaron Lennon came in
Everton substitutes Deulofeu and Funes Mori combined for what appeared to be the winner, before the late intervention of Terry.
MOAN OF THE MATCH
Surely the timing system in football can be more transparent and clear? We were told there would be seven minutes of stoppage time, only for a goal to come in the eighth
If seven minutes was not enough added time, why?
Arsenal v Chelsea (Barclays Premier League, Sunday, January 24)
Everton v Swansea (Barclays Premier League, Sunday, January 24)
Source : PA