Dragons also finished 15th in 2022-23 and last season.
In a statement Dragons said: “We thank Dai for his contribution, professionalism, and dedication to Dragons RFC throughout his time at the club and wish him and his family well for the future.”
Tiatia, who has been named interim head coach, will be “supported by the current coaching staff”.
Flanagan left his role as Scarlets backs coach to become Dragons head coach in June 2022, when Dean Ryan was director of rugby.
During Ryan’s protracted departure, which was finalised in January 2023, Flanagan took charge and remained in place from that point on.
Despite continued expressions of optimism that Dragons would become more competitive – and occasional wins to savour – they remain the least successful of the four professional sides.
With the autumn internationals prompting a break in URC fixtures, Dragons are next in action against Cardiff at Rodney Parade on Saturday, 30 November (17:15 GMT).
