In a short answer, yes.

When talking about "classical" music, there are two definitions- there's the Classical time period, which ran from 1750 (the death of J.S. Bach) to around 1820. There's also what most people consider "classical music," which encompasses the Baroque, Classical, Romantic, Contemporary, and Modern periods. You mentioned Bach, who is classified as a Baroque composer, so I'm assuming your question is directed at "classical music" as a genre, not as a time period. (Note: Some people refer to "classical music" as "Western art music," although this term is still debatable, as there are many non-Western "classical" composers of note, from places including eastern Europe, Japan, China, and more).In the Classical period, Turkish Janissary military music was very popular in Vienna for a time. After the Ottoman-Habsburg Wars between the Ottoman Empire and the Austrian Empire, as the name suggests, ended in the 18th century, the cross-cultural exchange that the war provided included the popularity of Janissary bands, elite mounted musicians whose music was mainly characterized by heavy percussion. We see this imitated in Beethoven's Turkish March. In Western music, brass was featured prominently, which imitated the militaristic themes. Mozart's Turkish March (aka "Rondo alla Turca") is scored for piano, but it follows that same percussive feel as the orchestrated pieces. Beethoven's "The Ruins of Athens" also takes inspiration from the Mevlevis sect of Islam's dance rituals. It should also be noted that classical guitar in Spain has roots in Islamic music, evident in pieces like Francisco Tarrega's "Recuerdos de la Alhambra" (Memories of the Alhambra).

In the Romantic period, which ran from around 1820 to the early twentieth century, there was also a prominence of Asian themes that emerged. Two famous examples of late Romantic music with Asian influence include Gustav Mahler's symphonic song cycle, "Das Lied von der Erde," and Puccini's opera "Turandot"."Das Lied" is based on text from a book of Chinese poems by Tang Dynasty poets including Li Bai, Meng Haoran, Wang Wei, and Qian Qui. It should be noted as well that Mahler added some of his own lines, possibly to elongate the poems, and that the poems do not translate exactly as they were originally written, as translating from Chinese to German provided some linguistic issues, as Chinese poetry is typically much more abstract. However, Mahler used many techniques common in Chinese music, including a heavy use of pentatonic scales. It's suggested that there are also some influences of Buddhism as well: the last movement of the piece, Der Abschied, ends with the German word ewig, meaning "forever," repeated over and over again, sustained by peaceful chords played by celesta, mandolin (added to the orchestra to give the piece a more Eastern feel, along with the tam-tam), and harps. This ending has been interpreted as a depiction of the Buddhist idea of Nirvana, with the repetition of the word ewig being seen as a sort of mantra. (Side note: if you ever get the chance to listen to Das Lied, you'll definitely get that impression!) Puccini's opera "Turandot," premiered in 1926, was originally based on a Persian epic called Haft Peykar by Nezami of Ganja, written in 1197. Although in the original story, the princess is Russian and much of the characters are Persian, the opera takes place in China and the characters are Chinese, with the libretto in Italian. However, it uses the popular Chinese song Mo Li Hua (Jasmine Flowers), from the Qing Dynasty.