When the sentence starts with the quote, follow it with a dash ( – ).
“Мы на пороге важнейшего открытия!” – сказал профессор.
“my na paROgye vazhNYEYsheva atKRYtiya” - skaZAL praFYEsar
“We are on the verge of a major discovery!” said the professor.
Rather than using quotes, you can just use dashes on either side of the quote:
– Мы на пороге важнейшего открытия! – сказал профессор.
- my na paROgye vazhNYEYsheva atKRYtiya! - skaZAL praFYEsar
“We are on the verge of a major discovery!” said the professor.
When the sentence ends with the quote, introduce the quote with a colon ( : ).
Профессор сказал: “Мы на пороге важнейшего открытия!”
praFYEsar skaZAL: “my na paROgye vazhNYEYsheva atKRYtiya!”
The professor said, “We are on the verge of a major discovery!”
Rather than using quotes, you can use a colon and present the quote on a new line, beginning with a dash:
Профессор сказал:
– Мы на пороге важнейшего открытия!praFYEsar skaZAL my na paROgye vazhNYEYsheva atKRYtiya
The professor said, “We are on the verge of a major discovery!”

Study Resource: How to punctuate dialogue in Russian?
Introducing the quote
Russian «Quotation Marks»
In Russian, you may see quotation marks written in a number of ways.
“...” → more common if using a US keyboard
«…» → more formal, common in published writing
„…“ → usually for quotations inside quotation marks
A direct quote should be in quotes, unless it is set off by dashes in the ways we saw illustrated above.
Punctuating the end of a direct quote
If the quote ends with an exclamation point or a question mark, that will always be inside the quotes (or the dashes).
«xyz!» “xyz!” „xyz!“ – xyz! –
«xyz?» “xyz?” „xyz?“ – xyz? –If the quote ends with a period or is not a complete sentence, the punctuation will be outside the quotes (or the dashes).
If the quote does not end the sentence, replace the period with a comma.
«xyz», – … “xyz”, – „xyz“, – – xyz, –
If the quote does end the sentence, use a period.
…: «xyz». …: “xyz”. …: „xyz“. – xyz.