I'm assuming you are asking for a word like "Prost!" (German) and "¡Salud!" (Spanish).
When you drink (can be alcohol or whatever) to celebrate, we often say "to have a toast to".
When you clang the mugs together like in a table for celebration, people often say "Toast!", "Cheers!", "Toast/Cheers to [the success]".
You and your friends are celebrating your birthday and are drinking beer around a table. A friend of yours can stand up, hold up a beer mug and say "Toast to [your] birthday!" and everyone clangs their mugs together.
This usually goes with alcohol but you can do the same thing with any other drink.
Since the OP edited their question to specify that they want something similar to " замачить " (Let's wet my new cam.), which is Russian, I'll add that you can say "Let's drink to [the event]!" to propose a celebration with alcohol.
For example: "Let's drink to Sam's birthday!" or "Let's have a drink to Sam's birthday", which are proposing that we celebrate Sam's birthday (the event) by drinking (most likely alcohol).