The deal carries an average annual value of $4.8 million.
Cirelli played in 68 games for Tampa Bay last season, during which he scored 16 goals and set career-highs in assists (28) as well as points (44).
The six-foot forward appeared in 25 Stanley Cup Playoff games for the Lightning, scoring three goals and nine points.
Tampa Bay drafted Cirelli in the third round, 72nd overall, at the 2015 NHL Draft.
News of his signing comes one day after the Lightning lost star forward Nikita Kucherov to what will be season-ending hip surgery.