The accomplished left-hander hammered 106, helping his side to a second innings total of 268 and a lead of 229 on Sunday.
Victoria went to stumps with 25 without loss in their second knock, needing another 205 runs on the final day Monday to snatch an outright win.
Bevan's century moved him to outright third on the list of century-makers in Sheffield Shield history.
Only current Test left-hander Darren Lehmann (37) and the legendary Don Bradman (36) have hit more.
Bevan's dismissal by Allan Wise (5-59) triggered a low-order collapse, with the last five wickets toppling for 20 runs. It was Wise's first five-wicket haul in top ranks.
Earlier, Bevan and New South Wales captain Mark Waugh put on 116 in a partnership that threatened to take the match away from Victoria. Former Test star Waugh, in possibly his last first-class game on the ground, made a fine 58 after being dropped twice.
The winner on what promises to be a tense final day will join Tasmania at the top of the Shield ladder.