The Magpies led by 20 points in the third quarter but fell behind early in the last term, before Dan McStay kicked the final goal of an 11.6 (72) to 10.11 (71) victory at the MCG.
Both sides had further chances to score but Gawn's snap at a boundary throw-in narrowly missed and a set shot by Koltyn Tholstrup fell short as the Demons fell agonisingly short.
Melbourne captain Gawn's shanked kick from the back pocket with 30 seconds left fell in the lap of Collingwood's Will Hoskin-Elliott, and the Pies iced the game.
Gawn and May were then involved in a heated conversation on the siren, with the captain pushing his teammate away.
Almost a year to the day since last season's injury nightmare, Demons star Christian Petracca returned to the same stage with 26 disposals and five clearances.
But it was some of Collingwood's stars who shone brightest as the ladder leaders made it five wins on the bounce and improved their record to 11-2 ahead of a mid-season bye.
Jeremy Howe (26 disposals) was brilliant in defence, Josh Daicos (34) stood up in the middle and McStay kicked two important goals.
Nick Daicos kicked one goal from 19 disposals as he dealt with a tight tag from rebadged Melbourne winger Ed Langdon.
The Demons (5-8) had their chances as Bayley Fritsch and Kysaiah Pickett kicked three goals each.
Pickett put Melbourne in front briefly in the final quarter and Clayton Oliver (30 disposals, nine clearances) worked hard.
The Demons dominated centre clearances (16-7) and overall clearances (39-29), and had a 54-45 advantage in forward entries, but couldn't make it count.
Langdon wore Daicos like a glove from the outset and their tussle was central to multiple spotfires throughout a tight first half.
Two Pickett goals bookended three straight Collingwood majors in the second quarter, his first a fortuitous hack off the deck from 45 metres.
But the Demons managed just 2.4 for the term - to Collingwood's 4.0 - despite dominating the inside-50 count (21-9).
The Magpies, who enjoyed an 18-11 advantage in free kicks in the first half, led by eight points at the main break after Tim Membrey slotted a set shot on the siren.
Fritsch twice trimmed the margin to a solitary point with brilliant snaps in the final term, either side of a Nick Daicos goal on the end of a 50m penalty.
And Pickett put Melbourne in front with another snap before McStay sent the Magpies ahead for the last time.
Collingwood's victory extended their recent dominance of Melbourne to 11 wins in 13 meetings.
