The Cumbrians, who won the Johnstone’s Paint Trophy at Wembley last week, had led with nine minutes left but Angelo Balanta levelled before Clayton came up with his late heroics.
Daniel Powell opened the scoring for the hosts after 13 minutes when he controlled Keanu Marsh-Brown’s cross from the left before calmly finishing.
Carlisle drew level just before the half-hour mark when Liam Noble’s low drive from 25 yards went underneath the dive of Dons goalkeeper David Martin.
The visitors then went ahead from the penalty spot after Sean O’Hanlon tripped Francois Zoko and James Berrett confidently converted.
But salvation arrived for the hosts nine minutes from time when Balanta was left completely unmarked to head in Peter Leven’s free-kick. And the turnaround was completed deep into added time when Clayton tapped in Dean Lewington’s low cross.