South Australia was dismissed for 346 late on day two after opener Weatherald notched a century alongside a steely contribution from Carey, and their tail also wagged.
Nick Winter then struck with the last ball of the day to have Tasmania 1/6, trailing by 86 after the first two days of play at Blundstone Arena.
Weatherald was out for 126 in Sunday's third session in Hobart, trapped lbw by Jackson Bird who finished the Tigers' best with 4-70.
Tailenders Joe Mennie (49 from 39 balls) and Nick Winter (42) threw the bat to give the Redbacks a 92-run first-innings lead.
Bird had earlier struck a double blow with the second new ball, removing Weatherald and Chadd Sayers for five.
It was Weatherald's second ton in two months against Tasmania, after scoring a career-best 198 in the sides' draw at the Adelaide Oval in early November.
He shared a counter-attacking 128-run stand with skipper Alex Carey, who fell for 73 to a sharp catch by Ben McDermott at second slip off the bowling of Riley Meredith.
It was another healthy long-form contribution from Australia's limited-overs wicketkeeper after his 143 last match against Western Australia.
Weatherald and Carey came together at 3-57 and helped SA build a solid total on a green-tinged wicket.
Tasmania were on Saturday dismissed for 254, headlined by Simon Milenko's maiden first-class ton.