Lacazette was second top scorer in Ligue 1 - the French top flight - last season with 28 goals.
He has scored 129 goals in 275 matches in all competitions since breaking into the first team in the 2009-10 campaign.
His league tally last season was bettered by only Paris St-Germain's Edinson Cavani, with 35.
Lacazette, who has won 11 caps for France, has been linked with several top clubs and looked set to join Atletico Madrid before their transfer ban was upheld by the Court of Arbitration for Sport.
Last week Arsenal chief executive Ivan Gazidis promised fans "top-quality" summer signings during a question-and-answer session.
Left-back Sead Kolasinac, who joined from Schalke, is Arsenal's only recruit since the end of last season.
Analysis
BBC Sport's David OrnsteinThe player Arsenal really wanted this summer was Alexandre Lacazette and sources there tell me his arrival is another signal that the Gunners are deadly serious when they say they want to become top-flight champions.
Although breaking the club transfer record does not guarantee anything, acquiring Lacazette proves Arsenal and Wenger can still attract top players, even without Champions League football.
Arsenal do have other targets but whether or not more signings are made is likely to depend on what happens with the likes of Alexis Sanchez, Mesut Ozil, Alex Oclade-Chamberlain and Jack Wilshere, all of whom are out of contract in a year's time and have been linked with moves away from the Emirates.
Arsenal's first-team squad is currently far too big and players need to leave before any more arrive.