The hosts led through Archie Thompson's 19th-minute opener, before ex-Victory man Sebastian Ryall levelled for the away side with 12 minutes to go in the first half. Extra-time looked to be a certainty after a tight second term in the elimination final before Brazilian playmaker Finkler struck in the dying seconds to snatch a dramatic win for Kevin Muscat's men.
It could prove to be Alessandro Del Piero's last professional game, with the Sky Blues' Italian great tipped to now call time on his playing career.
Victory now face either Western Sydney Wanderers or Brisbane Roar away from home for a place in the grand final, with that fixture depending on the outcome of the second elimination final between Central Coast and Adelaide United in Gosford on Saturday.
Victory enjoyed the bulk of possession early in the first half without creating a clear-cut opportunity. That changed in the 20th minute when Victory centre-back Pablo Contreras, dragged far out of position by Del Piero, dispossessed his fellow international marquee close to the half-way line.
Contreras played a square ball to James Troisi, who advanced on goal before slipping in Thompson. The veteran forward, so often Sydney's nemesis over the years, curled a good finish beyond Vedran Janjetovic and into the far corner of the net. It was an emotional goal for Thompson, who missed Victory's last match due to a family bereavement.
Conceding appeared to spark the visitors into life, and they began to enjoy far more of the ball, mounting a threat from a succession of set-pieces as the game approached the half-hour mark.
Despite being yet to genuinely test Nathan Coe in the Victory goal, the Sky Blue rally bore rich dividends in the 33rd minute. A headed clearance from Nick Ansell fell to Del Piero lurking outside the penalty area. The Italian passed to Ali Abbas in a wide left position and the midfielder's square ball back into the box found Ryall unmarked, affording the former Melbourne Victory man the simplest of chances to score against his old club for the third time in four games this season.
It had been something of a cagey affair until that point, but the equaliser proved to be the cue for an avalanche of chances at post ends. Nick Ansell and Mark Milligan were both inches away from heading Victory back into the lead from set-pieces, before Jason Geria had a speculative shot from long-range unwittingly blocked behind off a Sky Blue shirt. Thompson twice went close to adding to his tally, seeing one effort smothered by an onrushing Janjetovic and putting another wide of the far post.
It wasn't all Victory though, Sydney hitting the post through Del Piero and also testing Coe through Milos Dimitrijevic. The visitors made a statement of their attacking intent right from kick-off in the second half, Terry Antonis forcing Coe to turn the ball behind from a low, long-range shot. The Victory goalkeeper was then required to produce a brilliant reflex save to deny Nikola Petkovic a headed goal from the resulting corner.
Victory weathered that storm and the second half settled into a war of attrition, with both teams struggling to break the other down. With barely a shot on goal in the final 20 minutes, it looked for all the world as if the game would go to extra-time.
The hosts had other ideas however. A move started again by the unlikely figure of central defender Contreras saw Adama Traore win possession on the left side of the Sydney penalty area. The fullback looked up and picked out an inch-perfect lofted pass for Finkler, who supplied an ice-cool finish from the centre of the box to win the game.
Sydney's exit at the first hurdle of the play-offs is sure to cast more doubt on the future of coach Frank Farina, while Victory will now prepare for a tough assignment at either Suncorp Stadium against premiers Brisbane or a trip to Parramatta to take on last season's beaten grand finalists the Wanderers.
Melbourne Victory 2 (Thompson 19', Finkler 92')
Sydney FC 1 (Ryall 33')
20,802 @Etihad Stadium
