Hearts recovered from 2-0 and then 3-2 down to rescue a point in the Highlands, although their six-year wait for a win in Inverness continued.
Robbie Neilson's side showed commendable character and desire in a thrilling six-goal encounter with Inverness Caledonian Thistle.
Aberdeen's defeat by Celtic and St Johnstone's home loss against Partick Thistle meant a Hearts win would take them second in the Ladbrokes Premiership after Wednesday's missed opportunity at Kilmarnock. That chance was wasted again but it was difficult to criticise either side at full-time after such a pulsating affair.
David Raven and Lonsana Doumbouya put Inverness 2-0 up before Bjorn Johnsen reduced the score to 2-1 before half-time. Faycal Rherras equalised early in the second period but Aaron Doran immediately restored the hosts' lead. Arnaud Djoum had the last say with the equaliser.
Six years had lapsed since Hearts last won here, four since they even scored a goal. The visitors arrived desperate to exorcise the ghosts of Wednesday night at Kilmarnock, when they lost 2-0 with a particularly inept display. Johnsen and Jamie Walker came into the side, with strikers Tony Watt and Conor Sammon dropping out.
Inverness would have named the same starting line-up from their resounding 3-0 midweek win at Motherwell had goalkeeper Owain Fon Williams not pulled out through injury minutes before kick-off. Reserve keeper Cammy Mackay took his place.
Inverness threatened first with a counter-attack on six minutes. Carl Tremarco scampered past Liam Polworth on the overlap on the left side and his clipped cross was headed goalwards by Doumbouya. The striker could only hold his head in his hands, though, as his effort ball clipped the top of the crossbar.
An early change took place when Tremarco trudged off after ten minutes - presumably injured - to be replaced by Raven. Hearts forward Robbie Muirhead then tried an effort from 25 yards which spun narrowly wide of Mackay's right post. Raven normally plays on the right but five minutes after coming on at left-back he volleyed his team ahead.
A sublime passing move cut Hearts open as Liam Polworth clipped a through ball for Ross Draper to head down into Raven's path. The substitute lashed the ball first time with his right foot high into Jack Hamilton's top right corner to complete a superb team goal.
Hearts were more direct with their attacks and tried to quick balls forward to use the pace and physical presence of Johnsen through the middle. They were always susceptible to a breakaway, and on 32 minutes that's what put them 2-0 behind.
The hosts won possession in midfield and Iain Vigurs slipped a pass to Polworth on the run. He spotted Doumbouya peeling away to his right and rolled the ball through at precisely the right moment. The Frenchman took a touch before sending an elegant finish over the emerging Hamilton into the Hearts net from 16 yards.
Things looked bleak for the Edinburgh club at that moment but they responded quickly. Don Cowie's through pass found Johnsen on the run and he prodded the ball beyond Mackay to reduce the deficit to 2-1. Incredibly, it was Hearts' first goal at the Caledonian Stadium for four years.
The goal gave the visitors confidence for the second half and, seven minutes after the restart, they equalised. Rherras hoisted a cross in from the left and, as the Inverness defence hesitated with Johnsen lurking, the ball bounced straight through a clutter of bodies and into the net at the far post.
Just as Hearts looked to consolidate, another goal was surrendered. A long ball forward found Doran chasing Cowie, who was the last man between ball and goal. The Inverness winger nudged his opponent aside before stroking a calm finish low past Hamilton to restore the hosts' advantage.
Hearts now required to haul themselves back into the game for a second time. They pushed relentlessley for another equaliser and camped themselves in the Caledonian Thistle half. Eventually, Djoum found himself in space to the right of the penalty area for a vicious low drive which nestled in the corner of the net.
At 3-3, this was anyone's game. Mackay held an impromptu header from Djoum before Callum Paterson moved forward to join the Hearts attack for the final ten minutes. The pressure on the Inverness defence continued towards the last few moments, and Mackay reacted quickly to hold a shot from Cowie.
The final whistle sounded after three minutes of stoppage time and it was hard to argue that the 3-3 result was just about right. Neither team defended well, but the goals scored were worth the admission money.
Inverness CT (4-2-3-1): Mackay; McKay, Warren, Meekings, Tremarco; Tansey, Vigurs; Doran, Draper, Polworth; Doumbouya.
10: Raven for Tremarco
82: Mulraney for Doran
Unused subs: Horner, Boden, Sutherland.
Hearts (4-4-2): Hamilton; Paterson, Souttar, Rossi, Rherras; Djoum, Cowie, Kitchen, Walker; Johnsen, Muirhead.
54: Watt for Muirhead
60: Ozturk for Rherras
77: Sammon for Johnsen
Unused subs: Noring, Buaben, Oshaniwa, Nowak.
Referee: Craig Thomson.