After Sam Fanning (68) laid a stoic early platform, Jayden Goodwin (73no) and Cartwright (48no) helped lift WA to 2-220 at stumps on Saturday, 87 runs short of Tasmania's 307.
WA will look to erase the deficit early on Sunday before endeavouring to construct a match-winning lead.
Inside the last half-hour of play on Saturday, after a tough day of toil for the home side's bowlers, Mitchell Owen drew a thick edge from Cartwright, who was on 38, which was pouched low but cleanly by wicketkeeper Jake Doran.
The umpires got together before confirming, to Tasmania's shock, that the original not-out decision would stand.
Goodwin and Cartwright have combined for an unbeaten 91-run third-wicket stand, accelerating late after two sessions of attritional grind.
"We've got two guys who are set at the crease, so I think we're in a strong position," Fanning said.
"If we want to move the game forward, I think it's in our hands. We're set nicely."
Tasmania's lower order spent 17.3 overs poking around for 15 runs, after resuming on 7-302, before WA crawled to 0-23 at lunch in reply, the opening session reaping just 38 runs from 28.3 painstaking overs.
Lawrence Neil-Smith made six of 77 balls, while Kieran Elliott's 77-ball stay ended on 21 thanks to a marvellous one-handed slips catch from Cameron Gannon.
Teague Wyllie (22) was the only WA batter to fall before tea, trapped plumb lbw by left-arm orthodox spinner Matthew Kuhnemann.
Fanning dominated the 70-run first-wicket union with Wyllie and moved to his highest first-class score before chasing a wide half-volley from Bradley Hope, which he feathered to Doran.
"I see a pathway for us to victory - four overs to go before the new ball," Kuhnemann said.
"If we can take some wickets early tomorrow and put some pressure on, day three is moving day.
"I think the bowlers did a decent job, but tomorrow's a massive day for us."
