Alan Trouten's late goal for Brechin was not enough to overturn a 2-0 first-leg deficit, meaning that Alloa advance to the final.
Former Newcastle United and Cardiff city striker Michael Chopra and Kyle Benedictus had done the damage in the first leg, leaving League One Brechin needing a dramatic win if they were to make a return to Scottish football's second tier for the first time in 10 years.
Robert Thomson nearly pulled a goal back midway through the first half, but Chopra could have put the tie beyond reach with a chance of his own before the break.
Both teams had opportunities in the second half, before Trouten's goal with 10 minutes remaining gave Ray McKinnon's side hope.
A second Brechin goal would have forced extra time, but Alloa held firm in the final stages to book their place in the final and keep alive their hopes of playing in the Championship for a third successive season.