Sam Sodje - playing for the first time since completing a four-match ban - scored an own goal to hand Rovers the lead in the 50th minute.
Nicky Bailey pulled the Addicks level seven minutes later but neither side was able to find a winner.
Tranmere had failed to register a single shot on target in the first half, but went in front five minutes after the interval when Sodje inadvertently turned the ball into his own net after the ball had skimmed off Ian Thomas-Moore's head from a corner.
The home side responded well and skipper Bailey poked home an equaliser from close range after Deon Burton had headed a Grant Basey corner towards goal.
Ian Goodison headed a corner onto the bar for Tranmere, while Bailey curled a shot onto the post for the Addicks in stoppage-time. Rovers ended the game with 10 men after Zoumana Bakayogo picked up a second yellow card in the dying seconds.