Western Australia were bowled out minutes before stumps for 300, but Inglis held the innings together after the visitors' much-vaunted top-order slipped to 4-110.
Inglis was the penultimate wicket to fall, caught behind off Jackson Bird for 116 off just 134 balls.
At stumps, Tasmania were 1-5 with Caleb Jewell (1) failing to survive the three overs before the close.
Tasmania are one of four teams sharing second spot and eyeing a ticket to next month's final, with the decider almost certain to be hosted by runaway leaders Western Australia.
Tasmania's aggressive approach paid early dividends with the Shield's leading scorer, Cameron Bancroft, falling victim to a superb Tim Paine catch behind the stumps off the bowling of Bird for just one.
There were steady contributions in the top half of the WA order from Sam Whiteman (40), Teague Wyllie (28) and Hilton Cartwright (56).
WA went to tea after a rain-impacted middle session at a relatively precarious 5-184, but Inglis scored runs with regularity until his late dismissal.
Veteran Bird returned the best figures for the home side with 4-51.
33°C