The group, which backs Syria's government and which has previously hacked accounts of The New York Times, Agence France-Presse and other media organizations, published screen shots which it said backed its claims.
"Thanks to our operation, Twitter now blocks Obama's dangerous propaganda links," said a tweet from the Syrian group.
Both accounts appeared to be functioning normally some time after the claim. The White House did not immediately respond to an AFP query on the claim.
A Twitter spokesman said in a tweet that "The @BarackObama Twitter account was not compromised; their link shortener was."
The Syrian group also indicated it took over a Gmail account from Obama's campaign and a page from that website.
"We accessed many Obama campaign emails accounts to assess his terrorism capabilities. They are quite high," the group tweeted.
The SEA has made itself known in recent months, hacking the Twitter account of The Associated Press to put out a false tweet saying Obama had been hurt in two explosions at the White House.
SEA has also targeted the Twitter account of the AFP photo service, as well as social media at the BBC, Al-Jazeera and the Financial Times and Guardian newspapers.
The group alleges bias in the coverage of most media outlets over the unrest in Syria.