County took the lead through Chris Zebroski's fine half-volley from 20 yards after 16 minutes and Michael Flynn's penalty doubled their advantage shortly after the break.
But the Dons hit back quickly when Francomb's free-kick was deflected in by Kwesi Appiah before Francomb himself levelled up the scores with a deflected strike midway through the second half.
Zebroski, Newport's top scorer, bagged his 13th goal of an injury-hit season with a half-volley from just outside the area that gave goalkeeper Ross Worner little chance as it flew into the top corner of the net.
County were in charge and a rare away win looked to be a good bet when Christian Jolley was hauled down inside the box just 41 seconds after half-time. Flynn sent Worner the wrong way from the penalty spot.
But the hosts had other ideas and they hit back almost straight away when Appiah, the striker on loan from Crystal Palace, bundled the ball in at the far post from a Francomb free-kick with 49 minutes on the clock.
And the Dons were level after 69 minutes when Francomb unleashed a long range rocket that flew past Ian McLoughlin with the help of a deflection to earn the hosts a point.