Without doing a ton of research, Avenue Q has to be one of (if not the) earliest. It opened on July 31, 2003. Hairspray, La Cage aux Folles and 42nd Street all opened in August.
"The Pajama Game" opened on May 13, 1954 before winning the 1955 Tony, although dates for eligibility were different then, as the ceremony was presented in late March.