The second-seeded Razorbacks (25-7) led 55-35 early in the second half before missing 12 of their final 15 shots, a drought that allowed the Volunteers to rally to 69-65 with 3 ½ minutes left.
Madden (nine points) then made all of his attempts to start Arkansas' path toward victory at the foul line. The Razorbacks made seven of their next 10 attempts and 32 of 39 overall to advance to Saturday's semifinal against the South Carolina-Georgia winner.
Bobby Portis was 9 of 10 from the line en route to 26 points, including 18 in the first half. The newly selected SEC Player of the Year also grabbed 11 rebounds.
Senior guard Josh Richardson had team highs of 22 points and 10 rebounds for the Volunteers (16-16) before fouling out late.
What had shaped up to be the tournament's first rout ended up as another tight contest that Arkansas won thanks to strong free-throw shooting from nearly everyone.
Portis did his part at the line, but his dominant start gave Arkansas fans something to cheer by racing to a 10-0 lead in just over two minutes that included five by the sophomore forward. His 3-pointer prompted a timeout from Tennessee coach Donnie Tyndall in an effort to regroup.
Portis followed with another 3-pointer for an 18-3 lead before Devon Baulkman finally answered from long range for the Vols, whose challenge from that point was keeping the Razorbacks from breaking it wide open.
That was asking a lot against an Arkansas squad that got points from every starter except Harris, who contributed two early assists, but Tennessee kept it close for a little while in the first half by making free throws and getting a 3-pointer from Hubbs to get within 20-13.
Arkansas responded with a 19-6 run led by Portis, who had 18 first-half points. The spurt included a sequence where Harris' three-point play preceded Anthlon Bell's four-point play a minute later.
Bell made two more free throws for a 39-19 lead that looked safe until the Vols found their resolve and made it tense. Tennessee couldn't stop Arkansas' march to the line and missed some key chances down the stretch as well.
Tennessee: The Volunteers' bench outscored Arkansas 15-6.
Arkansas: Eleven of the Razorbacks' 16 first-half baskets were assisted. ... Arkansas missed all six 3-pointers in the second half and finished 4 of 16 from behind the arc.
Tennessee: Awaiting possible postseason bid.
Arkansas: Faces South Carolina-Georgia winner in SEC semifinal on Saturday.
Donnie Tyndall post-game audio