Abbott (4-50) did most of the damage for NSW, who sent Victoria in to bat at the MCG on Sunday and reduced their hosts to 9-259 at stumps.
Seven wickets fell in the final session, with Victoria losing 5-21 in the space of 12 overs shortly after tea.
But Tom Rogers (31 not out) and Fergus O'Neill (28) led a counter-attack against the new ball to ensure Victoria will bat again on day two.
Victorian opener Harris is in the mix to replace Steve Smith at the top of the order for Australia's Test series against India, starting in Perth next month.
The experienced left-hander made 143 and 52 in the opening Shield match of the season against Tasmania, but was out for 26 soon after lunch on Sunday.
Harris, who was hurt earlier when struck on the forearm, was caught down the leg side off returning Test paceman Mitchell Starc (1-35).
The 32-year-old hit just one boundary in his 89-ball innings and was the second wicket to fall after opening partner Ashley Chandrasinghe (11) was out cheaply to Aboott.
Peter Handscomb (64) and Campbell Kellaway (55) put on 101 runs for the third wicket before Jackson Bird (1-42) found the breakthrough after tea.
Handscomb hit 10 boundaries in his 112-ball innings before nicking Abbott to the safe hands of Smith at second slip.
Abbott also dismissed Sam Harper and Victorian skipper Will Sutherland for ducks in an impressive display, during which he sent down more overs (20) than any other NSW bowler.
Test spinner Nathan Lyon took 1-60 from his 19 overs.
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