It proved to be a sign of things to come for the busy Scunthorpe keeper, who denied Eoin Doyle on two occasions later in the half, including once when the striker was clean through on goal.
The hosts made another quick start after the break as Romuald Boco set up Doyle, who fired an effort narrowly past the upright.
Scunthorpe had a rare foray forward shortly before the hour mark when they stole possession in midfield and burst forward, but Paddy Madden's powerful low strike was wide of the target.
Chesterfield were soon back on the front foot, however, and the visitors needed Slocombe to come to their rescue again when he clawed away a Doyle cross.
The hosts threatened again with just over 10 minutes remaining of normal time, but Charlie Raglan's effort flew inches past the post.
Neither side could get a winner in the closing stages, and the opening period of extra time began slowly before Lee just about dealt with a Terry Hawkridge shot that took an awkward bounce.
The Spireites should have taken the lead just before half time in extra time when Jimmy Ryan moved into space and found Gnanduillet, who failed to keep his shot down when very well placed.
The hosts were ahead on their very next attack, however, as a low cross into the middle went all the way through to Clucas, who tapped home at the back post.
Scunthorpe went looking for a goal to keep them in the FA Cup and almost got it with 11 minutes remaining as Alex Davey saw a header blocked before his backheel was saved by Lee.
However, it was Chesterfield who got the crucial second goal as Darikwa set up a chance for Clucas, who bent a fine effort in to seal the victory.
The Spireites will now face Derby County in the fourth round later this month.