B737-800
Flight Crew Commercial
Situational Awareness
Deviation - Altitude Excursion From Assigned Altitude; Inflight Event / Encounter CFTT / CFIT
Flight Crew Became Reoriented; Flight Crew Returned To Clearance
Human Factors
I was the CA and PF.  We were on vectors and assigned altitudes with clearance to descend to fly the RNAV Y to Runway 4.  I used vertical speed to initiate descent.  We got low prior to DNNIS and received a call from LGA ATC.  We corrected to 1700 feet; VNAV PATH; and flew the remainder of the RNAV Y approach to Runway 4 for an uneventful full stop landing.Slow down sooner and; on a divert relocation flight; brief the whole approach prior to takeoff.
During the approach to LGA RWY 4; we were vectored direct to DNNIS for right dogleg to RWY 4; and cleared to descend out of 3000 FT to DNNIS at 1700 FT and to intercept the RNAV Y 4 course. We intercepted the course and continued descending below 1700 FT prior to DNNIS. LGA ATC gave us a low altitude call. We corrected up to the VNAV PATH and continued the approach to a normal landing. We could have pre-briefed the whole LGA approach while on the ground [prior to departure]. We also could have slowed down and configured earlier.
B737-800 flight crew reported descending below charted altitude of 1;700 FT on approach to LGA after a short flight.