Fred VanVleet and Cleanthony Early added 12 points each for the Shockers (7-0), who trailed by two at halftime but used hustle and physical play inside to wear down the Cougars.
Wichita State was clinging to a 68-62 lead when Baker was popped by BYU's Nate Austin, who was given a flagrent-2 foul and ejected from the game. Baker made both free throws with 45.3 seconds left, and then two more when he was fouled with just more than 30 seconds remaining.
The Cougars (5-2) were then hit with a technical foul on their bench for protesting a series of calls, and Baker tacked on two more free throws to put the game out of reach.
It was just the second in-season tournament that the Shockers have won since the 1963-64 season. They captured the Cancun Challenge before their Final Four run last season.
Matt Carlino had 21 points and Tyler Haws scored 17 for the Cougars, who missed a series of 3-pointers in the closing minutes that could have changed the outcome of the game.
As evenly matched as the teams were much of the night, both had long stretches of dominance in the first half before BYU emerged with a 36-34 lead at the break.
The Cougars, relying on a vexing zone defense, frustrated the Shockers over the first 10 minutes. Early was particularly perturbed, and coach Gregg Marshall decided to yank him from the game after a missed shot and a turnover on consecutive possessions.
BYU used a 13-2 run covering about 5 minutes to take a 21-8 lead.
Eventually, the Shockers started to have success shooting over top of the zone and the Cougars cooled off. Nick Wiggins curled in a 3-pointer, Tekele Cotton added one of his own and the scrappy Shockers roared on a 20-2 charge that ignited a partisan crowd inside the Sprint Center.
By the time Darius Carter finished it off with a bruising basket inside, the Shockers had taken a 26-23 lead - their first since the opening minute of the game.
It remained a back-and-forth game throughout the second half as the more physical Shockers tried to pound away inside and the suave, sweet-shooting Cougars peppered away from the outside.
The Shockers eventually took a 57-53 lead when VanVleet knocked down a 3-pointer from the corner with 5 1/2 minutes left, but Haws - held to four points in the first half - immediately took it to the rim and was fouled. He calmly made both free throws.
In fact, most of BYU's offense down the stretch came from the foul line, where the Cougars at one point scored six straight points. But they had also gone cold from the perimeter, while VanVleet scored on an assortment of 3-pointers and mid-range jumpers to extend Wichita State's lead.
Carter slashed to the rim with 1:30 left to make it 68-61, and after Carlino made one of two free throws for the Cougars, he missed a good 3-point look from right in front of his bench.
As chants of ''Let's go Shockers!'' echoed throughout the building, Baker was walloped by Austin at the other end of the floor. That began Baker's run of free throws that clinched the game.
BYU next plays Utah State at EnergySolutions Arena.