Ross County goalkeeper Scott Fox denied Hearts victory in the fifth minute of stoppage-time by saving Jamie Walker's penalty.
A frantic second half in Dingwall saw three goals and two penalties but ultimately it was Fox who saved a point for the hosts.
Youth coaches Jon Daly and Andy Kirk took charge of Hearts following Robbie Neilson's departure to MK Dons. They had to contend with a 1-0 half-time deficit due to Jay McEveley's goal for County, but Arnaud Djoum and Callum Paterson struck after the break to put the Edinburgh club 2-1 ahead.
Liam Boyce equalised in the 86th minute from the penalty spot, but Hearts wasted the opportunity to steal victory at the death. Walker's penalty was pushed to safety by Fox and both teams had to settle for a point each.
With the same starting line-up that beat Motherwell 3-0 and Rangers 2-0 in the last week, the visitors had arrived in the Highlands confident. A pre-match minute's silence marked the death of former County goalkeeper Benny Sutherland and the Chapecoense disaster.
Thereafter, Hearts put together a confident start to the match. Their first scoring chance arrived on 11 minutes when Paterson's downward header from Don Cowie's corner was cleared off the goal line by the County midfielder Tim Chow.
Bjorn Johnsen and Djoum both had attempts off target as the teams displayed some impressive flowing passing moves. One of those led to the clearest scoring opportunity of a very open first half.
A cross into the Hearts penalty area was headed out to Michael Gardyne, who forced it back in towards Boyce. His shot on the turn forced an instinctive parry from the Tynecastle goalkeeper Jack Hamilton, thus preserving the blank scoreline.
The deadlock lasted until two minutes from the interval. Hearts fell behind when they failed to defend a set-piece. Chris Routis' free-kick to the back post was headed down by Andrew Davies and McEveley, with his back to goal, showed good agility to hook the ball over his shoulder and high into the net.
County went for the jugular when the second half began and would have scored again but for Hamilton's brilliant goalkeeping. The Scotland squad member produced a stunning double save to repel two netbound efforts from Gardyne at the far post. Minutes later, he also stopped Boyce's shot.
The Highlanders' energy continued as they pressed Hearts all over the pitch. Johnsen managed to turn his marker, McEveley for a shot which Fox held low to his right. Hearts' American striker was fouled turning the same player moment later for a free-kick which led to the equaliser.
Robbie Muirhead's delivery was only partially cleared and Johnsen struck a first-time shot from around 14 yards. Djoum stuck his foot out in front of the keeper and stabbed the ball into the net. That imbued Hearts with belief at a vital time in the match and they struck again within minutes.
Djoum's cross from the right saw substitute Conor Sammon unsettle the home defence, and Paterson charged in to prod the ball past Fox. That sparked wild celebrations in the away end as the travelling support acclaimed the swift turnaround.
There was still plenty drama to unfold, however. Hamilton made another superb save to push Boyce's header for a corner. Then came the moment for Hearts to kill the game. On 78 minutes, Fox was dispossessed by Johnsen inside his own penalty area but, from a tight angle, the forward could only shoot into the side netting.
The away camp would rue that when County equalised. A cross to the back post was handled by Faycal Rherras and referee Willie Collum pointed to the penalty spot. Boyce stepped up for a convincing conversion from 12 yards to the game at 2-2.
Even then, Hearts should have left with three points rather than one. Walker was scythed down by Davies in the fifth minute of stoppage-time for game's second penalty. From the spot, Fox produced a crucial save low to his right to earn the Dingwall side a point and leave Hearts disappointed.
Ross County (4-4-1-1): Fox; Fraser, Davies, McEveley, Van der Weg; Dingwall, Chow, Routis, Gardyne; Boyce; Curran
65: Schalk for Curran
83: Woods for Chow
84: Dow for Dingwall
Unused subs: McCarey, Franks, Cikos, McShane.
Hearts (4-2-3-1): Hamilton; Paterson, Souttar, Rossi, Rherras; Cowie, Kitchen; Walker, Djoum, Muirhead; Johnsen.
64: Sammon for Muirhead
77: Ozturk for Rossi
84: Buaben for Cowie
Unused subs: Noring, L Smith, Currie, Watt.
Referee: Willie Collum.