First Officer reported late runway change and mismanagement of RNAV resulted in descending too low and a CFTT event.

Narrative

We were setup and planned for the visual backed up by the ILS XR approach into ZZZ and were descending via level change when ATC gave us a last minute change and switched us to the RNAV Y XR. We programmed the RNAV Y XR; and I engaged VNAV but didn't verify the path. We were approximately 1-3 miles SE of the ZZZZZ WP when we called the field with clear VFR visibility and were cleared for the visual approach to XR. As we descended thinking we were coupled to the VNAV PATH; we noticed we were pretty low visually to the airfield. I disengaged autopilot and auto throttle to stop the decent and the Captain told me to climb from 1900 ft MSL to 2700 ft MSL; the crossing alt at ZZZZZ1. We climbed to 2700 ft MSL; reconnected VNAV PATH; and continued the decent path to land visually with no further issues. Suggest: When using an approach to assist in a visual approach; ensure the approach is engaged and properly indicating the selected vertical and horizontal path; i.e. verify. Also; if ATC gives an approach change that feels too late to program; tell them which approach you're setup for and that you'd like to continue or get vectors to reset.