Connolly (90 off 115 balls) was the last man out in WA's first innings score of 347 on a green-tinged WACA Ground wicket.
At 20 years and 213 days, Connolly was on track to become the second youngest player to post a century in a Shield final.
The late Phillip Hughes (19 years, 106 days) is the youngest centurion in a Shield final, with current Tasmania captain Jordan Silk (20 years and 344 days) next in line.
Destiny appeared to be on Connolly's side when he was dropped on 86 after Brad Hope made a mess of a simple chance at deep backward point early on Friday.
But there was no getting past the nervous 90s for Connolly, whose knock was ended when he edged Iain Carlisle to slip with Beau Webster taking a sharp catch.
Carlisle (3-65) and spinner Jarrod Freeman (3-103) led the way for Tasmania with the ball, while Riley Meredith finished with 2-65.
WA's two selection gambles paid off handsomely.
D'Arcy Short, who was brought in to replace the concussed Cameron Bancroft as opener, scored 50 in a 110-run opening stand with Sam Whiteman (104).
But the bigger selection shock was the inclusion of Connolly for his debut.
Connolly was the hero for the Perth Scorchers in their BBL final win over Brisbane in 2023, striking an unbeaten 25 off 11 balls to get his team over the line with four balls to spare.
He brought a white-ball mindset into his Shield debut, cracking 12 fours and three sixes in his quick-fire knock.
But with the field spread waiting for a catch, Connolly's hopes of a century were brought undone when he edged to slip.
Connolly resumed on his overnight score of 73 with WA 8-325.
Cameron Gannon was the first to fall on Friday when he was caught behind down the leg side off the bowling of Meredith.
Riley Meredith gets the early breakthrough on day two and WA are nine down — cricket.com.au (@cricketcomau) #SheffieldShield pic.twitter.com/It1kS1FVEeMarch 22, 2024
Hope's dropped catch gave Connolly a chance to push on for his century, but his luck ran out a short time later.
YOUNGEST CENTURIONS IN A SHIELD FINAL
*19y 106d: Phillip Hughes
*20y 344d: Jordan Silk
*20y 360d: Martin Love
*21y 131d: Justin Langer
