All-rounder Beau Webster also continued his strong form late on day two, taking two India A wickets as the home side closed on a win.
The tourists were 5-73 at stumps on Friday, a lead of only 11 with two days' play left.
As fellow Test contenders Nathan McSweeney, Cam Bancroft and Sam Konstas all went cheaply, Harris made 74.
Australia A were dismissed at tea for 223, giving them a 62-run lead.
Khaleel Ahmed celebrates after dismissing young Test hopeful Sam Konstas cheaply. (Morgan Hancock/AAP PHOTOS)
Webster, who took 3-19 in India A's first innings, joined Nathan McAndrew in claiming two wickets late on day two.
Spinner Corey Rocchiccioli chimed in with a bizarre dismissal, bowling opener Lokesh Rahul for 10.
Rahul went to leave a delivery that bounced and was going down leg, only for the ball to somehow deflect off him and onto the stumps.
Australia's Test squad will be named soon after this match ends and all the interest lies in who opens the batting with Usman Khawaja against India.
McSweeney, the Australia A captain, was moved up the order to open with Harris, but he only made 14.
Bancroft's woes persisted when he was dismissed for three late on Thursday and Konstas was the first wicket to fall on Friday morning for the same score.
Immediately after lunch, the Indians were fuming when Harris defended a delivery from spinner Tanush Kotian and the ball deflected to first slip.
It was ruled not out and as Harris pointed to his pad, indicating there was no edge, Kotian glared at the umpire and may well face a dissent charge.
Tanush Kotian (l) thought he had dismissed Marcus Harris. (Morgan Hancock/AAP PHOTOS)
Putting a high price on his wicket, Harris faced 138 balls and hit five fours.
The opener was the seventh wicket to fall when he fended at a ball from Prasidh Krishna well outside off stump and was caught behind.
Harris, 32, has played 14 Tests, most recently the 2022 Sydney Ashes match.
Late-order resistance from Jimmy Peirson (30), McAndrew (26no) and Rocchiccioli (35 from 28 balls, with three fours and two sixes) gave Australia the first-innings lead.
Pace bowler Michael Neser, who was injured on day one, did not bat.
Krishna took 4-50, while Mukesh Kumar claimed three wickets and fellow opening bowler Khaleel Ahmed picked up two.
There were two more rain delays on day two, but Melbourne's notoriously fickle spring weather had cleared by mid-afternoon.
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