The Stags, rocked by full-back Tony Vaughan's red card for violent conduct after just 19 minutes, looked in deep trouble when Andy Robinson deservedly rifled Swansea into a 30th minute lead.
But six minutes into the second half, Liam Lawrence was tripped by Roberto Martinez after a mazy run into the area and got up to convert his 11th penalty of the season to earn a draw.
Elsewhere, bottom-of-the-table side Carlisle got another vital win by beating Scunthorpe 3-2.
Click here to read Footymad's reports on these games or click here to see how they affect the league table.