For an Italian visitor, this church sounds like a familiar presence. Indeed, the architect, Johann Bernhard Fischer von Erlach, having completed a large part of his training in Rome (1670-1685), could have "exported" a more imaginative and less conventional Baroque to this church, such as that of some Roman creations. by Borromini. Instead, for this church he adopted a more classical style, which is expressed, as well as in the dome, in the facade (which recalls the Pantheon), and in the two columns in front of the church (imitating the Trajan's Column).
It's reasonable to imagine that the client, who is directly the Holy Roman Emperor Charles VI, had some role in allowing the victory of the Fischer project, in the public tender that was held for the new church. However, the fact remains that a church like Karlskirche would be very plausible in Rome. The elliptical plan is almost the only element that links the church to the more "extreme" Baroque (as opposed to the numerous works by Fischer built in Salzburg): rather, according to some, it even seems to anticipate the neoclassical style.
That said, I would like to emphasize that the element that struck me most in this church is the close-up view of the frescoes in the dome. They, depicting the "Glory of San Carlo Borromeo", as well as the representation of some "Virtues", were executed by Johannes Michael Rottmayr. For some years, the restoration work on the dome has required the construction of very high scaffolding, which can be accessed by an elevator. Apparently the works are finished, but the scaffolding and the elevator have remained, and allow visitors a close view of the frescoes, which certainly was not foreseen by the painter. So many delightful details of the frescoes emerge, which enchant the visitor.