It was an improbable victory given that the Potters found themselves two goals down inside 47 minutes.
But the introduction of Matthew Etherington and Ricardo Fuller made all the difference with both players finding the target in a rousing comeback.
Stoke scored three times in the space of five minutes only for Ben Burgess to level matters.
It looked as though the tie was heading for extra time but Griffin's individual effort broke Blackpool hearts.
The Championship team shaded it in the first half and there was a scare for Stoke after 15 minutes when Charlie Adam spotted Burgess racing into the area. The delivery was inch perfect and Burgess' header came back off the post as the assistant referee raised his flag.
Adam was beginning to pull the strings in midfield and tried an audacious shot from the halfway line after 30 minutes that just cleared the bar.
Blackpool almost made the breakthrough five minutes later when Jay Emmanuel-Thomas got on the end of a Joe Martin cross, only to send his header over the top.
OpenerThe visitors eventually took the lead in the 40th minute with an emphatic finish from David Vaughan after he had been set up by Adam.
The Seasiders increased their lead two minutes into the second half when Billy Clarke got on the end of a corner from Vaughan to send a looping header beyond Steve Simonsen.
He wheeled away in delight after finding the target and Stoke now faced an uphill task to get back in the match.
However the hosts received a lifeline when they were awarded a penalty in the 56th minute after Tuncay was bundled over by Alex Baptiste and Ashley Asthma.
Liam Lawrence stepped up only to see his effort turned superbly away by Gilks, who had replaced Paul Rachubka in the Blackpool goal.
Stoke increased the pressure and Tuncay saw his 25-yard shot blocked by Gilks before Beattie fired over the bar from inside the area.
Tuncay should have pulled a goal back only to clear the bar when in a great position.
RewardedBut the home side got their reward in the 75th minute through Danny Higginbotham who got on the end of an Etherington corner to head home.
Three minutes later Etherington equalised with a crisp finish after Fuller had set up the opening.
The drama continued with Stoke then taking a 3-2 lead in the 80th minute when Fuller scored after linking up with Tuncay.
Blackpool though refused to lie down and Burgess pulled them level a minute later.
However, Stoke secured their passage into the fourth round deep in injury time when Griffin raced forward and his shot beat Gilks.