Miller turned aside all 14 shots he faced in a busy second period, enabling Tyler Myers to score the game-winner less than 1 minute into the third.
The Sabres had dropped their first two contests under Rolston, who took over behind the bench after longtime Coach Lindy Ruff was fired last week.
Cody Hodgson provided a goal and an assist, and Thomas Vanek had two assists for Buffalo.
The Lightning got a goal from Steven Stamkos for the sixth straight game but couldn't avoid their second consecutive loss.
Mathieu Garon made 19 saves for Tampa Bay.