Nuhiu caused problems again as he raided forward and was brought down on the edge of the box, but Caolan Lavery put his strike from the resulting free kick over the bar.
In the second half, Wednesday took the lead when Ryan Croasdale released the run of Marco Matias, who squared the ball into the path of Bus for a simple finish.
The Owls nearly doubled their advantage as Croasdale lifted the ball into the box for Jack Hunt, but he put his header narrowly over the bar, while at the other end Alfreton hit the bar through Dan Bradley.
Alfreton levelled the match in the 71st minute as they hit the Owls on the counter, and Niall Heaton headed home from close range.
Wednesday pushed for a winner, but were forced to settle for a draw in their pre-season opener.