The Stags went a man down when Krystian Pearce tripped Nicky Featherstone, and Paynter netted the resulting spot-kick.
Mansfield equalised when Colin Daniel rifled a half-volley into the top corner, but Paynter grabbed the winner from the spot after Mal Benning's foul.
The Stags, who announced first-team coach Micky Moore's departure prior to kick-off, have now not won in seven.
Two Billy Paynter penalties were enough to beat 10-man Mansfield and extend Hartlepool's unbeaten run to seven.