The Belgium international, 31, travelled to Istanbul this week to complete a medical after the two clubs agreed a fee reported at an initial €18m (£15.3m), which could rise by a further £1.7m in add-ons, paid in six equal instalments. His deal in Turkey also includes the option of a further year.
Trossard had one year left on his Arsenal contract, having joined from Brighton for a reported £21m plus add-ons in January 2023. He leaves having made 174 appearances in all competitions for the club, scoring 36 goals and registering 34 assists.
He was a key part of Mikel Arteta's squad during their run to last season's Premier League title, contributing eight goals and 11 assists in 50 appearances across all competitions, and started in the Champions League final defeat by Paris Saint-Germain on penalties.
Trossard also featured heavily for Belgium at this summer's World Cup, scoring twice and setting up two more as the Red Devils reached the quarter-finals. He came off the bench in their eventual 2-1 defeat to Spain in Los Angeles, having started the majority of Belgium's earlier matches in the tournament.
An Arsenal statement thanked Trossard for his contribution since joining from Brighton, while Besiktas are viewed as landing a significant coup in bringing an experienced, Premier League-winning forward to the Super Lig.
The Belgian scored a stoppage-time equaliser against Manchester City in the 2023 Community Shield, which Arsenal went on to win on penalties, and netted a decisive goal away to West Ham United last season that helped confirm their first top-flight title since 2004. Seventeen of his 27 Premier League goals for the club either levelled the scores or put Arsenal ahead, a record bettered only by Bukayo Saka during his time at the Emirates.
Arsenal are expected to reinvest the fee into further squad additions this summer, with Aston Villa's Morgan Rogers, Paris Saint-Germain's Bradley Barcola and Club Brugge's Christos Tzolis all having been linked as options to strengthen the left flank in Trossard's absence.
Gabriel Martinelli's own contract situation, with his deal also expiring in 2027, means Arsenal could enter next season with a very different look on that side of the pitch.