"You want to win the first game, especially at home, and that's what we talked about," Killorn said. "If you don't win the first one, it's such an uphill battle and it makes the series that much tougher. We knew it was going to be a physical game."
Kucherov scored twice, and Ben Bishop made 34 saves for Tampa Bay, which defeated Detroit in a seven-game first-round series last season.
Mike Green and Justin Abdelkader scored for Detroit, and Howard made 31 saves. Green had a chance to tie the game with less than a minute remaining in the third period when he controlled a loose puck in front of the net but his shot went over the net.
"I thought we had lots of moments of momentum and lots of moments of good hockey," Red Wings coach Jeff Blashill said. "We had some chances there at the end to tie it. At the end it was a tight game and we had an opportunity to [win] and we didn't do it."
Kucherov's first goal came at 6:23 of the first period. Killorn found Johnson with an outlet pass and Johnson set up Kucherov with a one-timer from the right circle that beat Howard.