John McMahon's men needed a win and Notts County to lose or dropped points from Crewe Alexandra to preserve their third-tier status, but could not pull of the great escape as they succumbed to a heartbreaking 2-1 home loss to Bradford City.
Matthew Pennington put Tranmere ahead in the seventh minute, but defender Junior Brown was dismissed for the hosts after 31 minutes.
And Tranmere's 10 men could not hang on as late goals from Jon Stead and Aaron McLean gave Bradford the points.
Meanwhile, Carlisle's demotion to the fourth level of English football was confirmed by a 3-0 loss at champions Wolves.
Graham Kavanagh's side required victory at Molineux and a Crewe defeat with a four-goal swing in goal difference to stay up, but Carlisle were well beaten as goals from Sam Ricketts, Michael Jacobs and Nouha Dicko helped Wolves reach 103 points, a new record for the third tier.
Swindon's defeat allowed Sheffield United to leapfrog them into seventh by beating Coventry City 2-1, while Peterborough United will approach the play-offs on the back of a 0-0 draw with Port Vale.
Elsewhere, Colchester United won 1-0 at Walsall, Crawley Town drew 1-1 with Bristol City and Gillingham were held to 1-1 stalemate by already-relegated Shrewsbury Town.
