Lampard, who scored England's match-winner against Brazil on Wednesday, got his 12th Chelsea goal of the season.
Wilshere hobbled off following a strong challenge from Alfred N'Diaye and Arsenal were reduced to 10 men when right-back Carl Jenkinson was dismissed for a second booking after fouling Stephane Sessegnon in the 62nd minute.
Despite Chelsea winning its first Champions League trophy and seventh FA Cup last season, manager Roberto Di Matteo has spent the offseason overhauling the club's attacking options ahead of the upcoming Premier League campaign.
Young Belgian striker Romelu Lukaku scored twice in the first half, Chelsea newcomers Eden Hazard and Marko Marin scored deflected goals, and the champions of Europe opened their American tour with a 4-2 win over the Seattle Sounders on Wednesday night.
Champions League finalists Chelsea have signed German international midfielder Marko Marin from Bundesliga side Werder Bremen, the Premier League club confirmed on Saturday.