Dunaway was drafted by the Green Bay Packers in the second round of the 1967 NFL Draft, the 41st overall pick. Dunaway spent the entire 1967 season on the Packers' taxi squad.John Maxymuk, "Taxi Squad" in George Bozeka, editor: The 1966 Green Bay Packers: Profiles of Vince Lombardi's Super Bowl I Champions (Jefferson, NC: McFarland and Company, 2016), p. 276. He began the 1968 NFL season with the Packers, but was cut after playing in only two games.Maxymuk, ibid. He was then picked up by the Atlanta Falcons and was with them for 8 games. However, he had no catches for either team in 1968. He was cut by the Falcons late in the 1969 preseason, and was briefly on the Washington Redskins taxi squad, before signing with the New York Giants. The Giants promoted him from their taxi squad chiefly to become the team's fourth punter in four seasons.Murray Chass, "DUNAWAY KICKING FOR NO. 1 BERTH", New York Times, December 11, 1969, p. 74. While he only was on the active roster with the Giants for the final three games of the 1969 season, his only marks in NFL statistics came there. He punted 13 times for a 38.2 yard average, caught two passes for a total of 37 yards (both in a 49-6 blowout win over the St. Louis Cardinals)ProFootballReference.com, "St. Louis Cardinals at New York Giants - December 7th, 1969", Available online: https://www.pro-football-reference.com/boxscores/196912070nyg.htm . Retrieved May 16, 2018. His lone carry was for 4 yards for the Giants, but it was an important play. It came on a successful fake punt that helped lead the Giants on a 4th quarter game-winning drive against the Pittsburgh Steelers.ProFootballReference.com. "Dave Dunaway". Available online: https://www.pro-football-reference.com/players/D/DunaDa20.htm . Retrieved May 16, 2018."Fake punt catches Steelers", New Castle (PA) News, December 15, 1969, p. 26. However, the Giants cut Dunaway in September, 1970, shortly before the season opener.Special to the New York Times, "Football Transactions". New York Times. September 2, 1970, p. 43. Instead, the Giants handed their punting duties to Bill Johnson, who had punted the four previous years for the minor league Orlando Panthers.Murray Chass, "Footloose Bill Johnson Captures Giants’ Fancy", New York Times, November 20, 1970, p. 48.
