The No. 4 Razorbacks advanced four consecutive base-runners with two outs in the 12th. Matt Reynolds started the rally with a two-out single before Bo Bigham, Matt Vinson and Wise drew consecutive walks to end the game.
Arkansas (17-2, 1-0 SEC) won its seventh straight game overall and eighth straight at home.
"We kept talking about in the dugout that this is one we had to find a way to win," Arkansas coach Dave Van Horn said.
"It's hard to lose a game like this because you play 12 innings and you battle, and you battle."
Arkansas answered in the bottom half of the inning when Vinson drew one of his three walks to leadoff the inning and scored on a Jacob Morris RBI double.
The Razorbacks stranded two runners in the ninth, however, when Dominic Ficociello lined out to Crimson Tide third baseman Brett Booth. Arkansas stranded 11 base-runners, while Alabama stranded 12.
Astin repeatedly worked out of jams in the extra innings, stranding Ben Moore at third base in the 10th and a pair of runners in the 12th.
"He did a great job," Van Horn said. "He pitched Tuesday and Wednesday, and we had him available to close.
"We didn't want to burn Barrett for the entire weekend, but we didn't want to lose."
Wise, who finished the game 2-for-4, also drew a bases-loaded walk in the bottom of the ninth inning to beat Northwestern State on Feb. 21.
"It helped a lot," Wise said of being in the situation previously. "You want to come in and be aggressive with the bases loaded. When you get to 3-2 you want to take ball four. I wanted to take ball four."
Arkansas used five pitchers in the win, but four – including Astin – will be available later in the weekend against the Crimson Tide (8-10, 0-1).
Both starters, Arkansas' DJ Baxendale and Alabama's Charley Sullivan, earned no-decisions by pitching 5 1/3 innings. For Baxendale, who entered the game with a 5-0 record, it was the first non-win in a start this season.
Alabama threatened in the first inning, but Vinson climbed the left field fence to rob Josh Rosecrans of a would-be two-run home run and end the top half-inning.
"We practiced that yesterday," Vinson said. "We went and practiced robbing home runs on that exact fence. I didn't even see it go in my glove. I can't believe I caught it."
"It was unbelievable," Van Horn said. "You knew that play was going to come into play the whole game. I don't know if I've ever seen a better one. When it left the bat I said, ‘It's gone.'
"I can't wait to watch it again."
Alabama eventually took the lead when Baxendale was called for a balk with a runner on third base in the third inning. The Razorbacks answered in the bottom half of the frame on a Tim Carver RBI single to score Wise from second base.
The two teams traded runs again in the eighth and ninth innings.
Alabama took a 2-1 lead on a wild pitch by Colby Suggs to score Jared Reaves, but the Razorbacks tied the game 2-2 when Ficociello scored on Bigham's two-out RBI double. Arkansas nearly took the lead on the hit, but Brian Anderson – pinch-running for Sam Bates – was thrown out at the plate to end the inning.
The two teams will meet again Saturday at 3:05 p.m. Sophomore right-hander Ryne Stanek (4-0, 1.12 ERA) will start for the Razorbacks, while Alabama freshman left-hander Taylor Guilbeau (1-1, 3.98) will get the nod on the mound.