Mr Podesta also suggested Republican nominee Donald Trump's campaign was given advance notice of the breach.
His remarks come after WikiLeaks published a trove of his hacked emails.
The top adviser to Mrs Clinton said the enquiry was part of a wider probe into hacks on the Democratic Party by groups with Russian ties.
"I've been involved in politics for nearly five decades," Mr Podesta told reporters aboard the Clinton campaign plane.
"This definitely is the first campaign that I've been involved with in which I've had to tangle with Russian intelligence agencies, who seem to be doing everything they can on behalf of our opponent."
The longtime aide to Mrs Clinton pointed to a tweet by Mr Trump's adviser Roger Stone in August that said Mr Podesta's "time in the barrel" was coming.
However, the tweet was posted shortly after WikiLeaks released thousands of hacked emails from other Democratic officials.
Mr Podesta also claimed Mr Stone has been in touch with WikiLeaks founder Julian Assange, but acknowledged evidence was "circumstantial".
Mr Stone told the AP news agency that Mr Podesta's claims were "categorically false" while WikiLeaks tweeted that it "has had no contact with Roger Stone".
Mr Podesta also speculated that the timing of the first WikiLeaks release on Friday was intended to distract from the explosive 2005 footage showing Mr Trump making obscene comments about groping women.
The Clinton campaign has not verified the authenticity of the leaked emails, adding that they could have been altered.
"The pattern is they hack, they leak truthful things, and then they build up to leaking documents that are either doctored or wholly fabricated,'' said Mrs Clinton's spokeswoman, Jennifer Palmieri.