The 32-year-old Labuschagne has not scored a Test century since July 2023, with Australia's Test number 3 averaging just 24.69 in his last 43 innings in the longest format.
He has been under pressure to retain his place in the XI after scores of 1, 31 and 4 in three innings in the most-recent home series against Bangladesh.
Australia chair of selectors George Bailey said: "We're not shying away from the fact that we'd like some more runs from Marn, that's clear.
"But we're also confident that he can still make a positive contribution to that squad. I think Marn will continue to prepare and refine how he will prepare for this series.
"There's potentially a simplification or an ability to focus purely on red-ball cricket for him in the fact that he won't be in the one-day series leading up to it. So that may be a benefit.
"He'll prepare incredibly well, and specifically for this tour. In that squad of 16, I'm confident that he can still make a positive contribution."
The 23-year-old Connolly will provide batting cover in the 16-man squad despite not having scored a hundred in 13 first-class matches, but did strike an ODI century while opening the batting in Bangladesh in June.
Josh Inglis is the other spare batter in the squad and is also the second-choice wicketkeeper behind Alex Carey, just as he was against Bangladesh.
"He's a good player, and one I think that's going to continue to develop across formats," Bailey said of Connolly. "Every time that we've had him on tour, he's a good learner. He's keen to expand his game.
"He's still so young in his career, but I think his ability to bat in a range of positions will continue to grow as he gets more and more experience."
Matt Renshaw has been retained as opener after replacing Jake Weatherald for the second Test against Bangladesh, while Matthew Kuhnemann is a surprise inclusion as a second spinner alongside Nathan Lyon.
Captain Pat Cummins, Mitchell Starc, Josh Hazlewood and Scott Boland are the fast bowlers selcted for the three-match series, with Michael Neser providing another seam option.
Cameron Green and Beau Webster are the two allrounders in the squad as they were against Bangladesh.
Australia Test squad: Pat Cummins (captain), Scott Boland, Alex Carey, Cooper Connolly, Cameron Green, Josh Hazlewood, Travis Head, Josh Inglis, Matthew Kuhnemann, Marnus Labuschagne, Nathan Lyon, Michael Neser, Matthew Renshaw, Steve Smith, Mitchell Starc, Beau Webster
Marnus Labuschagne has been axed from Australia ODI squad for the upcoming tours to Zimbabwe and South Africa, with Travis Head and Mitchell Marsh set for returns in the 50-over format.
The Proteas could be without Kagiso Rabada for their entire three-Test series against Australia after the fast bowler suffered a serious hamstring injury.
Pat Cummins has welcomed Australia's long-awaited return to South Africa, describing the Proteas as a "really strong side" as the visitors begin preparations for a three-Test series in October.
Bangladesh recovered from a disastrous start to fight back against Australia on a remarkable opening day of the second Test in Mackay in which 18 wickets tumbled.
