Anderson stopped 28 shots -- including 10 in the first period and 11 in the middle frame -- improving to 26-19-6 on the season with his 18th career whitewash.
Spezza's hat trick was the fourth of his career and Erik Karlsson also scored as the Senators won for only the second time in 10 games.
The Lightning's Mathieu Garon yielded three goals on 26 shots as Tampa Bay fell for the fifth time in seven contests.
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