Quick answer
С днём рожде́ния!
s dnyom rozhdéniya — literally “with the day of birth”
The options
С днём рожде́ния!Happy birthday!
Поздравля́ю!Congratulations!
Жела́ю сча́стья и здоро́вья!I wish you happiness and health!
С пра́здником!Happy holiday! — works for any celebration
С Но́вым го́дом!Happy New Year!
Two cases in four words
С takes the instrumental (с днём), and рожде́ния is the genitive after день: “with the day of birth”. The same pattern gives you С Но́вым го́дом and С пра́здником.
Never congratulate early
Wishing someone a happy birthday before the day is considered genuinely unlucky in Russia. Wait for the date — even by a few hours.
What you hear back
Спаси́бо большо́е!Thank you very much!
Спаси́бо, о́чень прия́тно.Thanks, that’s very kind.
Typical mistakes
С день рожде́ние!
С днём рожде́ния!
с + instrumental, then the genitive.
Congratulating the day before
Wait for the day
Russians treat early birthday wishes as bad luck.
Check yourself
Which is correct?
с + instrumental (днём) + genitive (рожде́ния).