The official accounts of Barack Obama, Joe Biden and Kanye West also requested donations in the cryptocurrency.
"Everyone is asking me to give back," a tweet from Mr Gates' account said. "You send $1,000, I send you back $2,000."
Twitter said it was a "co-ordinated" attack targeting its employees "with access to internal systems and tools".
"We know they [the hackers] used this access to take control of many highly-visible (including verified) accounts and Tweet on their behalf," the company said in a series of tweets.
Meanwhile, Twitter chief executive Jack Dorsey tweeted: "Tough day for us at Twitter. We all feel terrible this happened."
One cyber-security expert said that the breach could have been a lot worse in other circumstances.
"If you were to have this kind of incident take place in the middle of a crisis, where Twitter was being used to either communicate de-escalatory language or critical information to the public, and suddenly it's putting out the wrong messages from several verified status accounts - that could be seriously destabilising," Dr Alexi Drew from King's College London told the BBC.
Emergency response
Twitter earlier had to take the extraordinary step of stopping many verified accounts marked with blue ticks from tweeting altogether.
Password reset requests were also being denied and some other "account functions" disabled.
By 20:30 EDT (00:30 GMT Thursday) users with verified account started to be able to send tweets again, but Twitter said it was still working on a fix.
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