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Rodrigues was born in 1957 to Cuban immigrant parents in the Bronx, New York. After graduating from the University of Miami, he began his journalism career as a reporter at The Miami News before becoming a reporter at WTVJ-Channel 4. After 2½ years, he left Channel 4 for a reporting job at WPVI-TV in Philadelphia. In 1987, he returned to Miami to take up a weekend co-anchor and general assignment reporter at WPLG-Channel 10 and remained there for 11 years. Eliott appeared in the 1996 film "Up Close and Personal", playing the part of a television news reporter. He joined WFOR-TV (CBS4) in Miami in 1999, where he is currently the main news anchor and hosts the weekly public affairs program 4Sunday Morning. In January 2008 he was replaced by Antonio Mora as the main anchor; Eliot went to the Noon and 5:30 newscasts. In December 2012 Antonio Mora's contract was not renewed and Rodriguez went back to the 5:00, 6:00 and 11:00 weekday newscasts.
Rodriguez received two Emmy Awards for local news reporting in 2001 and 2004 and two Edward R. Murrow Awards. He was married to Univision anchor Maria Elena Salinas for 13 years before they divorced in 2007.