"We jumped out and switched a lot (on defense) in the second half, and I think it sparked a lot of offense," Sparks said.
"We knew it was going to be a tough game, so it was kind of what we expected. But we expected to come out with a little bit more intensity in the first half than we did."
Ball State (0-1) led twice in the second half. The Cardinals went ahead 35-33 with 17:12 left on a 3-pointer by Peyton Stovall, but Sparks keyed a 12-2 Kentucky run with his third 3-pointer of the game, giving the Wildcats the lead for good.
"He plays with a reckless abandon," Kentucky coach Tubby Smith said of Sparks, "which means he lays it on the line every time he's on the court."
Freshmen center Randolph Morris followed with a three-point play and later added a tip-in during the run, which ended with Kentucky up 45-37.
Ball State pulled within 45-40, but Alleyne led an 11-2 run with his three dunks, all in a span of three minutes. Two came on alley-oop passes, the third on a tip pass from Rajon Rondo.
That gave the Wildcats a 56-42 lead with 7:20 left, and they extended that margin to as many as 23 points. The Wildcats had only four turnovers in the second half.
"I don't think we necessarily stuck with the plan like we had in the first half," Ball State coach Tim Buckley said. "I think fatigue and the pressure contributed to that.
"We turned the ball over and we took tough shots."
Alleyne played in 18 of Kentucky's 32 games last season, averaging less than a point and just over three minutes per game. In two games this season, the sophomore played 16 minutes against Coppin State and 11 against Ball State.
He finished 5-of-6 from the field and helped the Wildcats shoot 60.7 percent from the field after halftime and 50.9 percent for the game.
"If I see a lane open to the basket, I'm going to it," Alleyne said. "If you stop me, you stop me, but if you don't, I'm going to the basket. I'm going to try to score any way I can."
Sparks went 7-of-10 from the field, including 4-of-6 from 3-point range. Morris and Chuck Hayes each added 10 points for Kentucky, while Hayes also had 11 rebounds.
Stovall led Ball State with 14 points, while Dennis Trammell added 13, going 3-of-3 from 3-point range. The Cardinals shot 36.1 percent from the field, the 34th straight game a Kentucky opponent has shot below 50 percent.
"Kentucky's defense is one of the best in the country," Ball State forward Skip Mills said. "We couldn't find the open shot against their defense."
Kentucky led by as many as eight points in the first half, but the Wildcats went almost five minutes without a basket and allowed Ball State to pull even at 29-29 by halftime.
Ball State entered the game with a history of pulling off upsets in its season opener. In 2001, the Cardinals played in the Maui Invitational in Hawaii to start the season and knocked off No. 3 Kansas 93-91 and No. 4 UCLA 91-73 on consecutive days before falling 83-71 to No. 1 Duke.