After an uneventful and scrappy first half, Allan Moore's men took the lead through inspirational skipper Mark McLaughlin, who stabbed a Dougie Imrie corner into the net.
They looked home and hosed when another Imrie set-piece wreaked havoc and Slovak Michal Habai lashed a fierce strike past goalkeeper Michael Andrews with just 10 minutes left to play.
But Liam Buchanan took advantage of two moments of Ton madness to score twice, with a placed effort and then a diving header on 90 minutes, and send the tie to extra-time.
Despite having the momentum, the additional 30 minutes proved a bridge too far for the part-timers. In the end, Ton's fitness told as they scored four more goals to sail into the second round.
Eighteen-year-old Ton debutant David McNeil was brought on with one minute left in normal time, and he got the ball rolling with an ice-cool finish on 93 minutes.
Fellow frontman Archie Campbell made it four when he pounced on a defensive mistake and raced clear before rounding exposed keeper Andrews and slotting into an empty net.
Campbell then saw a goal-bound effort pawed clear by defender Craig Johnstone - who was subsequently shown a red card - for a penalty in the 102nd minute.
He assumed responsibility and failed to beat Andrews with a side-footed spot-kick - but Imrie had followed up and swept under the advancing keeper to take Ton's tally to five.
But Campbell eventually bagged his brace two minutes from time when he rifled McNeil's cut-back high into the net.
Morton (4-4-2): Gaston; Taggart, Habai, McLaughlin, Fitzpatrick; Imrie, Hands, Bachirou (Fulton, 106), McKee (Wallace, 79); Campbell, Cham (McNeil, 89).
Subs not used: Page, Caraux. (gk),
Booked: Habai (72), McNeil (97), Taggart (113).
East Fife (4-4-2): Andrews; Durie, Dutot, Mbu, Johnstone; Willis (R. Stewart, 90), Brown, J. Stewart (Grieve, 73), Moosari; Clarke (Austin, 83), Buchanan.
Subs not used: McBride, Falconer.
Booked: Stewart (65)
Sent Off: Johnstone (102)
Referee: Craig Charleston