But the introduction of substitutes Ricardo Fuller and Matthew Etherington sparked a remarkable recovery, which was topped when Griffin powered into the left corner of the penalty are to drive a low shot into the far corner.
Both sides fielded severely depleted teams, with Stoke boss Tony Pulis leaving out his entire starting line-up from Saturday's draw at Bolton.
Ian Holloway also made eight changes to the 11 who kicked-off Blackpool's weekend victory at Nottingham Forest.
The first half was uneventful before the Seasiders went ahead five minutes before the break with a spectacular strike from Wales international Vaughan.
The former Crewe midfielder fastened onto a loose ball 25 yards out and fired a low shot which sped inside Steve Simonsen's near post.
Simonsen saved well from Clarke to concede a corner which Vaughan delivered for the diminutive Clarke to head high into the net from six yards.
Stoke finally woke up and should have pulled a goal back when Sanli Tuncay was fouled by Rob Edwards and Liam Lawrence's poor penalty was saved by Matt Gilks.
Gilks then produced a tremendous save to keep out a blistering free-kick by Tuncay before James Beattie fired an excellent chance over the top.
Pulis then brought on Fuller and Etherington and the latter soon laid the ball on a plate for Tuncay, who chipped over a gaping net.
The Premier League side drew level four minutes later with Etherington this time providing a close-range finish with his left foot after Tuncay headed on a cross from Lawrence.
And Stoke were ahead after just two more minutes when Fuller combined with Tuncay before placing a cool finish beyond the overworked Gilks.
Blackpool were still not finished and levelled 40 seconds later as Ben Burgess headed in superbly from a left-wing cross by Joe Martin.
Fuller, who along with Etherington had torn Blackpool's tiring defence to shreds, appeared to pull a hamstring shortly afterwards and Pulis opted to play the game out with 10 men rather than risk further injury to his star striker.
But Griffin's goal avoided the need for extra-time and Stoke can now concentrate on Saturday's visit of Manchester United.