The visitors finished six points ahead of their opponents in the Second Division and were quickly on top, going close in the 27th minute when Ryan McCord saw his effort crash into the post, with Kevin Moon blasting the rebound over.
The Wasps' domination continued and they took the lead six minutes before the break when McCord was brought down in the area. The striker saw his spot-kick saved by Michael Andrews, but he responded quickest to tuck the rebound home.
Alloa doubled their advantage 10 minutes after the restart when Moon curled in a superb 20-yard effort.
McCord then missed a chance to make it 3-0 when he dragged a shot wide, before Kevin Cawley twice went close to reducing Brechin's deficit.
But Alloa held on and go into Saturday's return leg as firm favourites.