Clarke's 87th-minute close-range header from Lee Holmes' free-kick sent the League One side through and prompted a full-time pitch invasion at the end of the match.
Tom Ince could have put Blackpool ahead when he found himself through on goal but his 18th-minute shot from just inside the area was off-target.
Angel Martinez went agonisingly close for Blackpool on 50 minutes after his header from Kirk Broadfoot's cross went inches wide.
Iain Hume then nearly put Preston ahead with his 63rd-minute shot that Matt Gilks did well to push around his post.
Michael Chopra had a golden chance to give Blackpool the lead but his 72nd-minute header bounced into the pitch and over the bar.
Blackpool pushed again and Bobby Grant's 84th-minute corner found Broadfoot but the defender headed over, and with Clarke capitalising moments later, the SkyBet Championship side crashed out.