Town had centre-back Anthony Charles sent off for bringing down Derek Riordan to prevent a scoring opportunity after 24 minutes, but survived until eight minutes after the break when Eaves side-footed Rovers in front from Eliot Richards' cross.
Richards then provided the corner for Kenneth to score the second at the far post after 57 minutes.
Adebayo Akinfenwa headed home a long throw from Ben Tozer to reduce Northampton's arrears after 75 minutes before Norburn wrapped up the points with six minutes remaining by nodding in substitute Lee Brown's cross.
Rovers were denied by the woodwork twice in the first half. Town goalkeeper Lee Nicholls did well to tip a fifth-minute volley from Eaves onto a post.
And after Charles' dismissal, Riordan curled the resulting free-kick over the wall and saw his effort clip the top of the crossbar.