Time-resolved CCD photometric observations of a dwarf nova EG Cnc were carried out during an outburst which occurred after a 19-year quiescent interval, since 1977 November. Our observations between 1996 December 2 and 14 revealed this outburst to be a definite superoutburst, eventually enabling a definite classification of this object as an SU UMa-type dwarf nova. This outburst was characterized by its long duration of more than two weeks and a large amplitude of at least 6.7 mag. The object showed two distinct types of superhumps: double-peaked modulations attributable to ``early superhumps'' with a period of 0.0582+/-0.00025 d in the earlier stage of the current outburst, followed by fully grown typical superhumps with a period of 0.0603+/-0.00006 d. We regard the period of 0.0582 d as being the orbital period of the object with a 3.7% superhump excess ratio. These characteristics suggest a strong resemblance to an extreme rare class of dwarf novae including WZ Sge, HV Vir, and AL Com with some other similarities between EG Cnc and the WZ Sge-type dwarf novae. The existence of double-peaked ``early superhumps'' in the early stage of superoutbursts may be a new defining criterion for the WZ Sge-type dwarf novae.