Super-sub Vassell struck twice in the space of four minutes to earn victory for Darren Ferguson's side.
He coolly steered in a Jon Taylor cut-back from close range after 81 minutes before beating Dons goalkeeper David Martin to an aerial duel following a Conor Washington pass four minutes later and tapping into an empty net.
The two teams had earlier exchanged strikes either side of the interval.
Posh loanee Kenneth McEvoy grabbed the first goal of his career with a wickedly deflected shot in the final minute of the first period.
But it was quickly cancelled out by Dele Alli just three minutes after the restart as he bundled home a rebound after a Will Grigg shot had been deflected into his path.
Vassell then put Posh in command but a nervy finish was assured when Dons substitute Daniel Powell burst through to reduce their arrears in the 87th minute.