McDermott and Doran were left to rescue the home side for the second time in as many days after 14 wickets fell on Tuesday at Bellerive.
After bowling Victoria out for 207 on day two, the Tasmanians collapsed to 4-14 before reaching 4-39 at stumps for a lead of 73.
McDermott had made 12 and Doran was on 14. They remain the only Tasmanians to reach double figures so far in this game.
Victoria were well-placed at 1-107 at lunch, before Mitch Owen took a Shield-best haul of 4-58 and Marcus Bean snared 3-32 to rip through the visitors' line-up.
"It's been a crazy game - lots of swings, proud of our efforts and showing some toughness" Owen said.
Marcus Harris and Ollie Peake put on 107 for the second wicket before Gabe Bell had Harris caught by Jordan Silk at second slip early in the middle session for 54.
Peake chased a ball from Bean outside off stump and chopped on for 50 without any addition to the score, leaving Victoria in sudden trouble at 3-116.
The next six batsmen reached double figures, but none scored more than 20 as Victoria squandered their solid start, losing 9-91.
Then came another momentum swing, as the Tasmanian top order gave Victoria's slips cordon plenty of catching practice. Victorian captain Will Sutherland took two wickets before McDermott and Doran guided Tasmania to stumps.
McDermott recovered from a back injury to top-score with 128 in the first innings, putting on 166 for the eighth wicket with Doran (73) after Tasmania were on the brink at 7-45.
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