Tampa Bay seemed to have what would have been its first victory of the season wrapped up, but Zajac fired a desperation slap shot that got past Mike Smith just before time expired.
Zach Parise and Jamie Langenbrunner then produced shootout goals to make the Devils a winner for the first time in three games.
Martin Brodeur stopped 41 shots in regulation before turning in his perfect performance in the shootout.
Tampa Bay has lost 12 straight games dating to last season. Two of its three losses this year have come in overtime.
Steven Stamkos scored twice for the Lightning, including a goal 1:52 into the third period that tied it at 2-2. Todd Fedoruk gave Tampa Bay the lead 4 minutes later and the Lightning protected that advantage until the game's final second.
Mike Smith took the loss, stopping 31-of-34 shots.
Copyright 2009 by United Press International.
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