Счастливого Нового года! (Schastlivogo Novogo goda!)

С новым годом!

S Novim godom! (in fact it's a "short" version of "Pozdravlyayu s Novim godom"="Congratulations with New Year(strict translation)" but in Russian language the verb can be absent in the sentense like that. Happy Birthday=S dnem Rojdenia(With a day of birth)))

S Novim Godom!