CE-560 Captain reported overshooting a crossing altitude resulting in an ATC low altitude alert. After a momentary CRM issue; flight crew corrected altitude and executed a safe landing.

Narrative

As the Captain I was the Pilot Not Flying on a short flight from ZZZ1 to ZZZ2. We were told to expect the RNAV RWY XX. The Copilot was the Pilot Flying but had never personally flown into ZZZ2 as flying pilot. I briefed him the approach was steeper than normal at 3.5 degrees and early aircraft configuration was key. I'm relatively new in the aircraft with approximately 60 hours in type; and while having a good grasp of the automation I'll acknowledge there is still a bit to be learned. I programmed the RNAV approach prior to being cleared for the approach.Approach cleared us 'direct to ZZZZZ cross at or above 4;000 ft.; cleared for the RNAV XX approach'. The Pilot Flying was in a VNAV descent and all seemed normal until we approached the ZZZZZ intersection. For whatever reason the aircraft automation didn't intercept the vertical path and just prior to ZZZZZ we both realized the descent path was going to cross us below 4;000 ft. prior to the fix. The Pilot Flying then turned off the autopilot to hand fly to capture the appropriate altitude. I went 'heads down' to check the sequencing of the FMS.Within a moment ATC issued us a low altitude alert and advised us to check our altitude. I immediately looked up from the FMS and saw the Pilot Flying hadn't leveled us at 4;000 ft. but had instead descended to an altitude between 3;600-3;700 ft.; which was the next fix crossing altitude. I will add; the weather was severe clear; all terrain was in sight and there was never a threat to the safety of the flight. The Pilot Flying quickly returned to the appropriate altitude for the approach. The rest of the approach was uneventful and the VNAV correctly captured shortly thereafter. Looking back I believe my main failure as the Captain and Pilot Not Flying; was my reaction to an unexpected VNAV situation. Instead of looking at the FMS; which my first instinct; I should have insured the situational awareness of the Pilot Flying was correct and he had positively corrected the flight path of the aircraft prior to looking down. Only then; should I have taken my attention away from the altitude and descent path to diagnose what the automation was doing.Overall; the safe outcome of the flight was never in question from the flight deck. Being in visual conditions added a safety buffer; no doubt. Going forward I will make it a priority to make sure any unexpected automation outcome takes a backseat to being in positive control of what the aircraft should be physically doing.