The Panthers (7-1) led by 8 with 1:46 to go.
Pittsburgh's Ashton Gibbs was called for a flagrant foul after elbowing Trae Golden, who was trying to foul him. Golden sank one of two foul shots, and Cameron Tatum hit a layup to cut the Panthers' lead to 5.
The Panthers missed the front end of two 1-and-1 free-throw opportunities, and a 3 by Golden with 11 seconds left cut Pittsburgh's lead to 58-56.
Dante Taylor missed another Panthers free throw. When Golden took the ball down the floor, Gibbs tied him up with the possession arrow in Pittsburgh's favor, setting up Johnson's baskets.
Johnson, a freshman playing in place of injured Travon Woodall, hit a layup on the break as he was fouled by Skylar McBee and sank the free throw.
Nasir Robinson had 16 points and 12 rebounds for the Panthers.
Tatum led Tennessee (3-4) with 13 points and 10 rebounds. Golden and Jeronne Maymon each added 12 points.
The game was a rematch of last year's Big East/SEC Invitational game at Pittsburgh's Consol Energy Center. The 11th-ranked Vols won that one 83-76, handing the third-ranked Panthers their first non-conference loss in their home city in nearly six years.
This installment featured far different lineups but was possibly more physical and definitely more competitive.
The teams traded the lead eight times in the first half alone with two ties.
Three times Pittsburgh led by 3 points, the last on a long jumper by Ashton Gibbs with 8:26 in the first half.
Tennessee scored the next three baskets to swing the 3-point lead to its advantage. Tatum hit a 3-pointer in front of the Panthers' bench to give the Vols a 26-20 lead with 4:22 before halftime, their largest lead of the half.
After a 2-minute scoring drought, Johnson sank a 3 for Pittsburgh, which scored the last three baskets of the first half to take a 27-26 lead into halftime.
Despite tough defense on both sides, both teams muscled their way inside for most of their baskets. The Vols got 18 of their first-half points in the paint, while the Panthers scored 12.
The first few minutes of the second half was more of the same until Tennessee's defense began to break down.
That allowed Pittsburgh to get three easy shots close to the basket in a 2-minute stretch. From there, the Panthers took control of the offensive boards and turned a 17-7 offensive rebounding advantage into 18 total second-chance points.
Big East's preseason player of the year Ashton Gibbs, who struggled against the Vols last season, warmed up as well, hitting two jumpers and a 3-point shot in a short stretch. Gibbs followed that up with a long field goal with 5:01 left that gave Pittsburgh a 53-45 lead.
The Panthers are on a five-game winning streak since losing to Long Beach State on Nov. 16.
The Vols have dropped four of their last five, including losses to No. 3 Duke and No. 22 Memphis.