Dallas Eakins began his rise to the ranks of the NHL by playing Junior B hockey in Peterborough back in 1983-84 at the age of 16. In 1984-85, he joined the OHL's Peterborough Petes and scored eight points as a major junior rookie. After scoring 22 points in 1985-86, he scored 14 points in 1986-87. In 1987-88, Eakins scored eleven goals and 38 points with the Petes.
In 1988-89, he turned pro with the Baltimore Skipjacks and scored ten points as an AHL rookie. Eakins was drafted 208th overall in the 1985 Entry Draft by the Washington Capitals. After a year in the Capitals' organization, he signed as a free agent with the Winnipeg Jets on October 17, 1989. It took three long years of toiling in the minors, but eventually he made his NHL debut in 1992-93 with 14 games and two assists with the Jets.
On July 8, 1993, Eakins signed as a free agent with the expansion Florida Panthers. He was traded to the St. Louis Blues for a fourth-round draft pick in 1995. He played 16 games with the Blues and was then claimed on waivers by the Winnipeg Jets on March 20, appearing in just two games that season. In 1996-97, he moved with the franchise to Phoenix and played four games at the NHL level. Once again, Eakins was traded on February 6 with Mike Eastwood to the New York Rangers for Jayson More. He got into the lineup for seven games with New York.
In July 1997, Eakins signed as a free agent with the Florida Panthers. That year, he played a career-high 23 games at the NHL level. The following summer he signed as a free agent with the Toronto Maple Leafs, but like his other stops around the league, his stay in Toronto was brief, lasting all of 18 games.
On August 12, 1999, Eakins signed as a free agent with the New York Islanders. He played just two games with the Islanders and was traded to the Chicago Blackhawks, but did not play any games with the club.
Eakins signed a free-agent deal with the Calgary Flames in July 2000, where he would be inserted into the lineup for 17 games. He later played with the Chicago Wolves of the AHL before signing as a free-agent with the Vancouver Canucks in the summer of 2003.