On-loan Liverpool goalkeeper Peter Gulacsi spilled a sixth-minute free-kick by Johnnie Jackson on his debut, presenting an easy opener to Scott Wagstaff early on in Saturday's League One clash at Prenton Park.
It could have been 2-0 to the visitors three minutes later but Joe Anyinsah fired wide when well set.
As Tranmere rallied, Lucas Akins header over and John Welsh volleyed just wide, while the hardworking Lee Martin twice went close for Charlton.
But Tranmere levelled after 52 minutes when a Dale Jennings header was blocked by Therry Racon, only for the loose ball to fall to Showunmi to ram home.
Showunmi then went close with a free-kick, while Paul Benson's header nearly won it for Charlton with 13 minutes remaining.