Staveley got off to a fast start finiding the back of the net after just seventy-five seconds.
They doubled their advantage just before the hour mark, but it took a discussion between the referee and his linesman before the goal was given.
Teversal fought back and in a well-worked move pulled a goal back through Mark Parkes on 76 minutes, after Danny Pearson sent the ball in from out wide, a deft touch by Ashley Lewis to set the ball up for Parkes to hammer home.
Try as they might, Teversal could find a leveller and Staveley went through to the next round to face Dronfield Town. Teversal will now face Pinxton FC on Wednesday evening in a second game.