Orlando cut the lead to one in the waning seconds, but came up empty at the foul line attempting to tie it.
Utah snapped a two-game losing streak with its fifth consecutive regular-season victory over Orlando. The Jazz played without point guard Mo Williams, who is out indefinitely with a thumb injury.
The loss also ended the Magic's three-game home winning streak.
Orlando led 73-71 entering the fourth quarter, but the Jazz scored the first seven points of the period and took a 78-73 lead with just over 10 minutes to play.
Trailing 92-90, Orlando rebounded a miss by Millsap with less than a minute to play and called a timeout. The Magic ran a set play for Afflalo, but his 12-foot jumper was blocked by Hayward and recovered by the Jazz.
Foye was then fouled and hit a pair of free throws to push the lead back to four.
Redick hit a 3-pointer out of a timeout to cut it back to one, and Millsap hit only 1 of 2 at the free throw line after getting fouled with 10.1 seconds remaining.
Jameer Nelson pushed the ball upcourt and found Gustavo Ayon underneath, who was immediately fouled by Foye. Ayon missed the first free throw and his intentional miss on the second was rebounded by Tinsley, who hit a pair from the stripe for the final margin.
With Davis sitting for the second straight game since spraining his left shoulder last week, Magic coach Jacque Vaughn started the second-year big man Ayon in his place -- his first start of the season. Davis is out indefinitely, with his return dependent on the progress he makes during rehabilitation.
Vaughn had gone with rookie Andrew Nicholson in Davis' place during the Magic's loss to Toronto on Friday, but struggled from the field in just 19 minutes of action. Ayon came on as a reserve in that game and posted a double-double.
At 6-foot-10, Ayon gave the Magic a size advantage over the 6-8 Millsap. Ayon finished with nine points and 12 rebounds, but Millsap clearly won the night.
The Jazz also had to adjust their starting lineup Sunday, inserting the veteran Tinsley at point guard for Williams, who sprained his right thumb in Saturday's loss at Miami. Williams' return in unknown after an MRI on Sunday revealed damage to the thumb. He will have it re-evaluated back in Salt Lake City.
The Magic took a 47-46 advantage into halftime, but led by as many as 16 in the opening 24 minutes.
Afflalo had 13 points in the first half to lead Orlando, which outscored Utah 16-14 in the paint and 13-8 on second-chance points.
Utah battled back, though, connecting on 4 of 9 from the 3-point line.
Millsap led the Jazz with 10 first-half points.
Notes: Sunday was the fifth game this season Williams has missed due to injury. He missed one game in early November with a strained right abductor and three games late last month with a sprained right foot. ... The Magic teamed with Chase and "Operation Homefront" at halftime to surprise Army Sergeant First Class Paul Hiltibidal and his family with a new, mortgage-free home and a $2,500 check to help provide holiday gifts for their four children. While fighting in Afghanistan, Hiltibidal was injured and retired from the Army earlier this year.