Chasetown FC will need a replay after being held in the FA Cup.
The Scholars should have had a penalty after 15 minutes when Jake Lovatt had his shirt tugged as he was about to shoot for goal, but the referee waved the appeals away.
Wednesfield then shocked their higher league opponents with the opening goal after 25 minutes. Kevin Nickle cut in from the right wing and fired a left footer past Curtis Pond.
A minute before the break, Lovatt received the ball on the edge of the box and drilled a first time effort wide of the post.
Chasetown levelled nine minutes into the second half. A ball was played into Lovatt who smashed a volley past Reiss Jeavons.
Wednesfield were reduced to ten men when Fabian Facey was sent off for violent conduct.
The Scholars came from behind and another through ball put George Cater clear of the last defender and he finished confidently.
But Wednesfield levelled with three minutes of normal time remaining when Nickle got on the end of a cross to head past Pond.