The fourth tier side, who joined the league in 1921 but have long been eclipsed by local powerhouses Liverpool and Everton, had lost their previous four matches and parted company with manager Micky Adams last weekend.
Tranmere reached the League Cup final in 2000 under player-manager John Aldridge, losing to Leicester City. They also missed out three times in successive playoffs for promotion to the Premier League.
Former FA chief executive and ex-Rovers player Mark Palios took over the club with his wife last year.
"For all of the fans and for myself it feels devastating today," Palios told Sky Sports television. "But tomorrow the sun will come up. It's not a disaster tomorrow.
"The fans away today gave me a humbling experience and something that makes me even more determined to make sure we get back as quick as we can."
Cheltenham were also relegated to the minor league for the first time since 1999.
(Reporting by Alan Baldwin, editing by Tony Jimenez)
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