A last-gasp try from Christ Tshiunza earned Exeter a one-point victory after Harlequins fought back from 31-7 down to lead going into the final minute.
Tries from Harry Williams, Stuart Townsend, Tshiunza and Harvey Skinner put the home side seemingly in control at half-time.
But four second-half tries without reply saw the pendulum swing in the away side's favour.
The lead changed hands twice more before Tshiunza sealed the victory.
With the clock in the red, the 20-year-old flanker crashed over in the left corner after the Chiefs had moved the ball downfield.
Even then the drama was not over as, with the suspicion of a forward pass by Stuart Hogg in an earlier phase, the TMO was called but, after seeing no clear evidence, referee Ian Tempest stuck with his on-field decision to send Sandy Park into raptures.
Exeter went into the game with two wins from two at the start of the new Premiership season and wasted no time in trying to add a third, with prop Williams touching down over after only three minutes.
After a Luke Northmore reply was chalked off by the TMO, Townsend powered over before Tshiunza coasted home through three challenges for a 21-0 lead.
Cadan Murley hit back for Harlequins before Skinner capitalised on a drop from Lennox Anyanwu to charge through and beat the chase to secure a four-try bonus point before half-time.
The away side responded in style after the break, with Tyrone Green and Murley both setting each other up to finish quick, decisive Quins moves in a two-minute blitz.
With Quins' Jack Walker and replacement hooker George Head both injured, referee Tempest declared uncontested scrums for the remainder of the match and Quins immediately capitalised, working the ball out from the scrum for Northmore to score his first of two tries.
His second stunned the home crowd as Harlequins took the lead for the first time.
Going behind appeared to finally kick the hosts back into gear as they kicked for the corner and drove forward, before substitute Pat Schickerling powered over to regain the lead.
Exeter's Ian Whitten was given a yellow card for slowing the game down with Murley baring down on the line for a hat-trick.
And the away side were ahead again moments later as Joe Marchant collected a pinpoint kick from Marcus Smith, expertly stopping himself and the ball from going into touch before nonchalantly securing the latest try of a crazy game and put Quins 42-38 up.
But the Chiefs refused to give up and won with the last play of the game through Tshiunza.
Exeter Chiefs: J Simmonds; Nowell, Slade, Whitten, Woodburn; Skinner, Townsend; Hepburn, Cowan-Dickie (capt.), Williams; Van Heerden, Gray; Ewers, Tshiunza, Capstick
Replacements: Yeandle, Kenny, Schickerling, Jenkins, S Simmonds, Maunder, O'Loughlin, Hogg
Sin-bin: Whitten (70)
Harlequins: Green; Marchant, Northmore, Anyanwu, Murley; Smith, Gjaltema; Garcia Botta, Walker (capt.), Louw; Hammond, Herbst; Kenningham, Evans, White
Replacements: Head, Baxter, Kerrod, Lamb, Jurevicius, Care, Allan, David
Referee: Ian Tempest