Cklamovski took over the side after being Ange Postecoglou's assistant with Yokohama F. Marinos last season when they won the title.
The 41-year-old also previously worked with Postecoglou at Melbourne Victory and with the Socceroos.
Shimizu produced their most impressive performance of the season so far, and the victory sees them rise off the bottom of the table, leapfrogging Kashima Antlers.
The match was delayed for several minutes with S-Pulse leading 2-0 due to wet weather.
S-Pulse are now just three points off Postecoglou's Yokohama who continued their early-season struggles with a 3-1 loss to Consadole Sapporo earlier on Sunday.
Elsewhere, Andres Iniesta created a number of chances but to no avail as Vissel Kobe lost 2-0 to Gamba Osaka.
