After losing the toss in Hobart and being sent in, Victoria were all out for 222 in the third session with quick Riley Meredith (4-53) the chief destroyer.
The hosts reached 4-52 in reply at stumps, with Charlie Wakim (nine) and nightwatchman Matt Kuhnemann (three) at the crease.
Harper, the leading run scorer in the Shield this season, earlier top scored for Victoria with a rapid 93 off 115 deliveries after opening the batting and dodging some first-over carnage.
Victoria were left reeling at 2-0 after veteran quick Jackson Bird (3-46) got rid of Campbell Kellaway and Blake Macdonald for ducks.
Harper, who was the Big Bash League player of the season, launched a salvage mission with Matt Short (40) who ended up caught in the gully off Meredith.
Harper was struck on the helmet on 81 by a Meredith bouncer but responded in style with back-to-back sixes.
But Meredith had the final laugh two overs later, when Harper skied a cut to a running Beau Webster in the deep.
Harper was coming off 119 and an unbeaten 141 in Victoria's last-start win over Western Australia that secured them a spot in the Shield final.
Victorian spinner Todd Murphy chipped in with what could prove a valuable 27 not out from just 23 balls at No.11.
Tasmania's top order was then rattled by Victoria's quicks, with Fergus O'Neill (1-8 from nine overs) and Sam Elliott (2-13) among the wickets.
Fourth-placed Tasmania and all other states bar last-placed Western Australia can make the final.
The Tigers are without Jake Weatherald in the penultimate Shield round after the Test opener suffered back spasms at training on Wednesday.
He is aiming to be back for Tasmania's final-round game against Queensland beginning March 14.
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