Melbourne's Carl Valeri had the first opening of the match but his effort was saved by Ben Kennedy, before Archie Thompson fired over from inside the penalty area after being found by Daniel Georgievski.
The home side continued to dominate, with Fahid Ben Khalfallah forcing another stop out of Kennedy, but Victory should have been ahead when Thompson flicked the ball wide from close range.
Jets had been quiet, but shortly before half time, they came agonisingly close to edging in front through Enver Alivodic though his header somehow missed the far post.
However, 10 minutes into the second half, the visitors went ahead through Montano, who capitalised on indecision from Melbourne inside the box before firing the ball inside Lawrence Thomas's near post.
As the game approached a conclusion, Finker drew a fine save out of Kennedy and Kosta Barbarouses fired the ball wide from 12 yards, but Newcastle somehow managed to hang on to record a three valuable points at the bottom of the standings.