It was the latest public spat over the transfer of power since Mauricio Macri narrowly beat Ms Fernandez's chosen successor.
She boycotted Mr Macri's inauguration two weeks ago in a row over the venue.
Ms Fernandez had led Argentina since 2007, taking over the presidency from her husband Nestor after elections.
Nestor Kirchner, who died in 2010, came to power in 2003.
The administration of Mauricio Macri now uses the Twitter handle @CasaRosada - named after the iconic pink presidential palace in Buenos Aires.
One of the first messages posted on the account reads: "Today the #CasaRosada finds us united. There's change in the air and we're ready to grow."
Less than 24 hours after being unveiled, the new official site was showing more than 100,000 followers.
The previous official account @CasaRosadaAR now describes itself as a "Twitter tribute" to the presidencies of Ms Fernandez and her late husband, Nestor Kirchner - and "not official from 10 December 2015". It currently shows 326,000 followers.
Twitter removed the blue badge marking it as a verified account on 10 December, AFP news agency reports.
Mr Macri's victory was the first in more than a decade for Argentina's centre-right opposition and ended the Peronist Party's 12-year rule.
Argentina's new president has unveiled a new official Twitter account after outgoing leader Cristina Fernandez de Kirchner decided to keep the old one.