The hosts had the better of the early chances, but Fahid Ben Khalfallah and Finkler both missed the target with their shots in the opening 10 minutes.
Khalfallah continued to find his way through for Melbourne and kept the ball himself a few minutes later, but Kennedy made a diving save to deny the midfielder, while another effort struck the crossbar.
Jacob Pepper tried to put the Victory under pressure against the run of play, unleashing a powerful strike from outside the box, but Coe parried it away.
The Melbourne keeper had to remain alert late on as the Jets pressed for a late winner, but Coe prevented Griffiths getting his head to a corner before denying the striker again from a Zenon Caravella cross.
Andrew Hoole tried to find the top corner with a curling effort inside the final 10 minutes, but his finish was a fraction wide.
The miss proved costly as the hosts finally broke the deadlock a few minutes later as Finkler fired in a free kick from just outside the area after Besart Berisha had been brought down.
The result moves Victory back up to second in the table, four points behind leaders Perth Glory, while Jets stay ninth.