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".
Media playback is not supported on this device
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.

Allegri, 48, took charge in July 2014 and signed a new deal until 2017 after leading the Old Lady to the league title and the Coppa Italia.
Having successfully defended their league crown, Juventus have announced the former AC Milan boss will now remain until 30 June 2018.
Juventus face AC Milan in the Coppa Italia final on 21 May.
They are 12 points clear of nearest rivals Napoli with two games remaining in Serie A.
Juventus defeated Manchester City twice in the group stage of this season's Champions League before being knocked out by Bayern Munich in the round of 16.

Media playback is not supported on this device
The 23-year-old Scot, who holds five outdoor and indoor British records, ran a personal best four minutes 19.12 seconds over the distance last year.
Budd's mark, set in Switzerland in 1985, is 4:17.57.
"It would be brilliant one month out from the World Championships," Muir told BBC Sport.
"To have another British record, and in the London Stadium again, would be fantastic. Fingers crossed I will be able to do something."
The world record - set by Russian Svetlana Masterkova in 1996 - stands at 4:12.56.
At last year's Anniversary Games, Muir broke the British 1500m record that had stood since Dame Kelly Holmes' gold medal-winning run in the 2004 Olympic final in Athens.
She expects her return to bring back happy memories.
"The main thing was just how loud the crowd was," said Muir.
"I am usually very focused on my race but I was just aware as I came into the final 200m of the crowd getting louder and louder. It was brilliant."
With the Anniversary Games just four weeks before Muir plans to take on a 1500m and 5,000m double at the same venue in the World Championships, it will also be a chance to maximise home advantage.
"It is really really helpful to know where the call-room, the warm-up areas, the kit drop-off and the toilets are," she added.
"Knowing that set-up is one of athletes' biggest worries going into a major championship.
"You want to have everything ready and not have to worry about anything else apart from doing the best you can on the track."
Alongside leading Britain's next generation of elite athletes, Muir is also training to become a vet.
Juggling studying and sport dates back beyond the London 2012 Games, when she organised a placement on a dairy farm so she would finish inspecting the morning milking round in time to watch the Olympic action at home in Kinross.
This year she has already squeezed in experience with the Dogs Trust and various local practices around winning 1500m and 3,000m gold at the European Indoors. The rest of the season is dedicated solely to athletics.
"After the disappointment of finishing seventh in Rio, we knew how important the indoor season was for me in the build-up to London," she added.
"We wanted to do really well there. We ticked that box pretty well and now everything is building towards London."
Media playback is not supported on this device
Arsenal captain Mikel Arteta and fellow midfielder Tomas Rosicky will leave the Premier League club when their contracts expire this summer.
[NEXT_CONCEPT]
Juventus boss Massimiliano Allegri has extended his contract with the Serie A champions until the summer of 2018.
[NEXT_CONCEPT]
Laura Muir will attempt to break Zola Budd's 32-year-old British mile record at the London Anniversary Games on 9 July.