There is already an excellent answer provided by the Thane of Cawdor (great username) but I wanted to pitch in from a different angle. I do a PhD in History of Art but specifically on the use of music in a particular 19th-century movement, when popular concert halls and orchestral venues became a large-scale phenomenon for the first time. By large-scale I mean, anyone could go and music took on a different role in Victorian London. There was one painter, raised in industrial Birmingham, never heard a proper orchestra before, who - upon his first experience - thought he was witnessing the 'Day of Creation'. He was also deeply religious at the time, but it struck me how the experience of music at that time and place (Victorian era) generated a quasi-religious experience.

It is also very true that orchestral performances were commonly loud, raucous, far from the revered silence-attitude we're used to today. That was first introduced in the early- to mid-19th century following the popularity of classical instrumental music coming from composers like Beethoven, Haydn, Schubert, Mozart, etc. The shift from social entertainment to independent artwork, to be appreciated as is, led to a shift from noise to revered silence. New concert halls were being constructed (St James's Hall, 1858; Handel Festivals and Saturday Concerts at the Crystal Palace from 1855 onwards; the Hallé Orchestra, 1857; London Philharmonic Society, 1813; Royal Albert Hall, 1878) were built to be such shrines of culture and classical music, dedicated entirely to this new style of appreciating music. I've noticed that, among those artists who were already religiously inclined, their responses to orchestral performances - and I mean large-scale ones, of c. 100 musicians - were decidedly religious. Specifically performances of Beethoven's symphonies were considered to be transcendent, mystical, and divine, an approach modelled on the original German Romantic response to Beethoven's music (specifically E.T.A. Hoffmann in 1810).

There was an easy connection between the impact of a large-scale orchestra and overwhelming sensory experiences, but not everyone interpreted it that way. My research focuses heavily on musical immersion and the artists I work on were mostly interested in colour-music, tonal relations, formal harmony, and the impact on the senses. That being said, these are artists, not every person on the street (though most of these artists did start off as kids in poor industrial families). There was definitely a large sense of wonder that many could see as religious, but others saw as purely sensuous. I think for many religion did provide the rhetoric to interpret these sensations, at least in the nineteenth century (many responses to Wagner's music; H.R. Haweis in London; revivals of Early Music in Oxford student societies in the 1840s and 1850s).

In addition to the answers already provided, there is also evidence of upperclass members of society having "borderline religious" experiences. For example, in a biography of Rev. Robert Hall, an English Baptist minister who lived 1764-1831, it's written:

Mr. Hall it runs was present in Westminister Abbey at Handel's commemoration [an event that took place in 1784 to commemorate the twenty-fifth anniversary of the death of George Frideric Handel, a Baroque composer]. The King, George III, and his family were in attendance. At one part of the performance of the Messiah (the "Hallelujah chorus") the King stood up, a signal for the whole audience to rise; he was shedding tears. Nothing, said Robert Hall, had ever affected him more strongly; it seemed like a great act of national assent to the fundamental truths of religion.

As you can see, even nobility who had heard orchestral works many times before could be affected in such an overwhelming manner.