We present observations of three core-collapse supernovae (CCSNe) in elliptical hosts, detected by the Zwicky Transient Facility Bright Transient Survey (BTS). SN 2019ape is a SN Ic that exploded in the main body of a typical elliptical galaxy. Its properties are consistent with an explosion of a regular SN Ic progenitor. A secondary g-band light-curve peak could indicate interaction of the ejecta with circumstellar material (CSM). An Ha-emitting source at the explosion site suggests a residual local star formation origin. SN 2018fsh and SN 2020uik are SNe II which exploded in the outskirts of elliptical galaxies. SN 2020uik shows typical spectra for SNe II, while SN 2018fsh shows a boxy nebular Ha profile, a signature of CSM interaction. We combine these 3 SNe with 7 events from the literature and analyze their hosts as a sample. We present multi-wavelength photometry of the hosts, and compare this to archival photometry of all BTS hosts. Using the spectroscopically complete BTS, we conclude that 0.3%(+0.3)(-0.1) of all CCSNe occur in elliptical galaxies. We derive star formation rates and stellar masses for the host galaxies and compare them to the properties of other SN hosts. We show that CCSNe in ellipticals have larger physical separations from their hosts compared to SNe Ia in elliptical galaxies, and discuss implications for star-forming activity in elliptical galaxies.