Leeds, knocked out of the FA Cup by Conference side Histon last Sunday, went ahead from an 11th-minute corner. Fraser Richardson flicked on a cross by Robert Snodgrass for Enoch Showunmi to beat Danny Coyne from eight yards.
Tranmere drew level in the 17th minute. United goalkeeper David Lucas missed his punch from an Andy Taylor free-kick and Antony Kay headed in from 12 yards.
A fine save by Coyne prevented Luciano Becchio regaining the advantage for the visitors after 25 minutes.
Tranmere capitalised three minutes after the break as Moore beat Lucas with a speculative 20-yard shot with the ball going in off the right-hand post.
Leeds thought they had levelled after 61 minutes but a Becchio header was ruled out for a push on Kay.