Tampa Bay goaltender Ben Bishop earned his first play-off shutout with 31 saves, also a personal play-off high, and the Lightning won the final two games to take the series 4-3.
Coburn, limited to just four games in the regular season due to a foot injury, suffered after the Lightning acquired him from the Philadelphia Flyers, broke a scoreless tie early in the third period.
He one-timed a long shot on an assist from right winger Ryan Callahan, beating Red Wings goalie Petr Mrazek to the top right corner of the net.
Lightning added an empty-net goal from defenseman Anton Stralman with 1:19 left to secure the win and advance to face the Montreal Canadiens in an Eastern Conference semifinal series.
Tampa Bay won all five meetings with Montreal in the regular season after being swept by the Canadiens in the opening round last year.
Detroit outshot the Lightning 23-12 in the first two periods, but could not pass Bishop.