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.
It added that "significant steps" were taken to limit access to such internal systems and tools while the company's investigation was ongoing.
Meanwhile, Twitter CEO Jack Dorsey tweeted: "Tough day for us at Twitter. We all feel terrible this happened."
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 accounts started to be able to send tweets again, but Twitter said it was still working on a fix.