Russia says it launched a "special military operation" to disarm its neighbor. Ukraine and Western countries say the invasion was an unprovoked war of aggression.
A separate conflict has played out online as Western sanctions have isolated much of Russia's economy from world trade.
"We've seen sustained intent from Russia to disrupt Ukrainian government and military systems. We've seen what looks like some spillover of activity affecting surrounding countries," Government Communications Headquarters (GCHQ) chief Jeremy Fleming said in a speech in Canberra on Wednesday.
"And we've certainly seen indications which suggest Russia's cyber actors are looking for targets in the countries that oppose their actions."
Russia has consistently denied accusations of mounting cyberattacks on Western targets. Its defense ministry did not immediately respond to a written request from Reuters for comment on Fleming's remarks.
Fleming dismissed recent commentary which expressed surprise that Russians had not deployed a major cyberattack alongside their invasion, an analysis that he said: "misses the point."
But "that's not to say that we haven't seen cyber in this conflict. We have – and lots of it," he added.
Dozens of Ukrainian and Russian websites have also been taken offline during the conflict by legions of "hacktivist" volunteer hackers on both sides.
Russian hackers have recently attempted to penetrate the networks of NATO and the militaries of some eastern European countries, Google's Threat Analysis Group said in a report published on Wednesday.
The report did not say which militaries had been targeted in what Google described as "credential phishing campaigns" launched by a Russian-based group called Coldriver, or Callisto.
"These campaigns were sent using newly created Gmail accounts to non-Google accounts, so the success rate of these campaigns is unknown," the report said.
Hackers are targeting European charity groups working to help Ukrainian refugees fleeing the country following Russia's invasion, a senior Ukrainian cybersecurity official said on Wednesday.
Victor Zhora, the deputy chief of Ukraine's State Service of Special Communications and Information Protection, did not provide much further detail.
His comments follow a report published earlier this month that alleged that unidentified hackers were trying to disrupt European government personnel involved in the managing the massive flow of refugees out of the country.
"We suppose that this is another proof of the spread of cyberwar to NATO countries," Zhora told reporters.