The IceCube Neutrino Observatory has discovered diffuse astrophysical neutrinos, however, despite finding evidence for neutrino emission from sources such as TXS 0506+056, the sources of the diffuse astrophysical flux remain to be seen. Fast Radio Bursts (FRBs) are a class of galactic and extra-galactic astrophysical objects that are known to produce periodic and non-periodic transient radio bursts. FRBs could be responsible for a portion of the diffuse astrophysical neutrino flux. IceCube has previously performed two searches for neutrino emission from FRBs with events that originate from charged current muon neutrino interactions. We present a search for neutrinos that are spatially and temporally coincident with 22 unique, non-repeating FRBs and one repeating FRB (FRB121102) using a different event selection of neutrino-induced cascades. There cascades originate from all-flavor neutral-current interactions as well as electron and tau neutrino charged-current interactions. Compared to previous searches, this event selection allows us to probe afterglow timescales with longer durations due to the low background rate. Since no statistically significant correlation of neutrinos with FRBs was observed, we set upper limits on the time-integrated neutrino flux emitted by FRBs for afterglow timescales that range from subseconds to weeks. These are the first limits to be set on neutrino emission from FRBs with neutrino-induced cascades.