After 21 wickets fell during the first two days at the Junction Oval in Melbourne, Kellaway (147) and former Australia opener Marcus Harris (101 not out) took advantage of perfect batting conditions on Thursday.
Victoria reached 5-330 before declaring just shy of stumps on day three with a lead of 355, just after Harris brought up his 33rd first-class hundred.
Fergus O'Neill (1-10) claimed the key wicket of Test aspirant Jake Weatherald (12) as the Tigers slumped to 1-28, requiring 327 for an unlikely win on Friday.
Coming off a middling start to the season, Kellaway found the form that has gained him a reputation as one of Australian cricket's brightest young talents.
The 22-year-old opener batted sensibly, lifting Victoria out of a precarious position at 4-91 with Harris in support.
After moving to No.3 this season after spending most of his career as an opener, Harris didn't come walk to the crease until No.6 in this innings due to the Vics sending Mitch Perry (five) in as a nightwatchman.
Like Kellaway, Harris hadn't fired yet this season, but was able to steady at the crease and cruise past 8000 Shield runs during his innings.
Tasmania's bowlers found it tough going, with speedster Riley Meredith (2-87) the only one to take multiple wickets.
Meredith finally got Kellaway out late in the day, ending a 195-run fifth-wicket stand with Harris.
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