The Gamecocks (49-18) took the lead in the bottom of the seventh inning on Barrett Astin's two-out, bases-loaded walk to Adam Matthews.
South Carolina will play Arizona in the best-of-three finals beginning Sunday. The Gamecocks are trying to become the first team to win three titles in a row since Southern California won five straight from 1970-74.
The Gamecocks beat Arkansas twice in three CWS meetings over five days after winning two of three against the Razorbacks during the Southeastern Conference regular season.
Arkansas had ended South Carolina's record 22-game win streak in the NCAA tournament with a 2-1 win on Monday. The Gamecocks forced a second bracket championship game with their 2-0 win on Thursday.
As expected, Friday's rematch came down to pitching.
The Gamecocks erased a 2-0 deficit in the fifth, with Arkansas starter DJ Baxendale walking Christian Walker with the bases loaded to force in the tying run. Baxendale had a career-high five walks.
Matthews looked ready to swing on a 3-2 pitch from Astin in the seventh, but he held up as the slider dropped low and Joey Pankake trotted home for the go-ahead run.
The run ended the Arkansas bullpen's streak of 26 scoreless innings and came on the ninth walk issued by the Razorbacks.
Colby Suggs (7-1) took the loss for Arkansas (46-22).
Webb, who relieved Colby Holmes after the Hogs' first two batters reached in the third, allowed two hits in four innings.
Price (5-4), who gave up one hit and struck out five, pitched three innings to earn his record fifth career win in the CWS. Of his 48 pitches, 37 were strikes.
Price was spectacular for the second year in a row in the bracket final. Last year he entered a 2-2 game against Virginia and worked 5 2-3 scoreless innings in a 3-2, 13-inning victory.