Arteta, 34, joined the club in 2011 and made 150 appearances, scoring 17 goals.
Rosicky, 35, signed in 2006 and was given an award for 246 appearances and 10 years of service before the Gunners' final-day win over Aston Villa.
Manager Arsene Wenger described the pair as "two exceptional players and influences in the dressing room".
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Spaniard Arteta and Rosicky of the Czech Republic leave Arsenal with two FA Cup medals, won in 2014 and 2015.
Arteta, who signed from Everton for £10m, thought he had scored on his farewell but his late strike in the 4-0 win over Villa on Sunday was given as a Mark Bunn own goal.
He was then seen crying after the final whistle.
In his match programme notes, Arteta said that failing to win the Premier League was "not good enough for this club".
Tottenham's 5-1 defeat by Newcastle meant the Gunners leapfrogged their north London rivals to finish second in the table, but they have not won the title since 2004.
Arsenal captain Mikel Arteta and fellow midfielder Tomas Rosicky will leave the Premier League club when their contracts expire this summer.