Jay Stein had a good shot parried clear by Alloa goalkeeper Allan Creer.
But Andy Brown rolled an attempt across the face of the Arbroath goal and Stevenson volleyed over before scoring.
The former Real Mallorca man volleyed the winner home before Robbie Raeside missed a great chance at the other end as Alloa progressed 2-1 on aggregate.
Alloa manager Allan Maitland:
"I think the game was pretty scrappy and we scored with the one moment of inspiration, which came from a great strike by Jamie Stevenson. "We have a young team and I hope they can relax more.
"We managed it for 20 minutes on Wednesday but couldn't today.
"If we are going to beat Berwick in the final, we will have to play better."
Alloa: Creer, Anthony Stevenson, Ovenstone, Townsley, Bolochoweckjy, McColligan, James Stevenson, Grant (McGeown 46), Andy Brown (Ross Hamilton 46), McLeod, Sloan.
Subs Not Used: Nicolson, Forest, Ferguson.
Booked: Ovenstone, Bolochoweckjy, McColligan, McGeown.
Goal: James Stevenson 69.
Arbroath: Inglis, Rennie, McCulloch, Raeside, Dobbins, Bishop, Miller (Watson 74), McMullen, Reilly (Taylor 66), Swankie, Stein.
Subs Not Used: Cook, Voight, Cargill.
Booked: McCulloch, Dobbins.
Att: 1,318
Ref: M McCurry
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