The 30-year-old Italy international tapped in after 22 minutes to match Argentina forward Gonzalo Higuain's record of 36 goals, set while playing for Napoli in the 2015-16 season.
Immobile also wins the European Golden Shoe for the continent's top scorer this term, having overtaken Bayern Munich striker Robert Lewandowski's tally of 34 in midweek.
Fourteen of his goals were from the penalty spot, as Lazio finished the season in fourth place, earning a return to the Champions League.
It is the third time Immobile has won the title of best scorer in Italy after 2013-2014, when he scored 22 goals with Torino, and in 2017-2018 when his 29 goals for Lazio saw him share the honour with Inter Milan's Mauro Icardi.
Cristiano Ronaldo missd out on winning a first 'Capocannoniere' award for top scorer in the Italian top flight, as he was rested for Juventus's final game, a 3-1 defeat to Roma.
The 35-year-old five-time Ballon d'Or winner finished second best with 31 goals.