The 27-year-old has signed a four-year contract after passing a medical in Stuttgart today.
He will wear the number 25 shirt for his new club when he joins them for pre-season training at the start of next month, making the same journey as his former Hannover team-mate Konstantin Rausch.
"I'm really pleased that we have been able to attract Moa to us," said Stuttgart's director of sport Fredi Bobic. "He is a player with international experience and extraordinary ability.
"We are now more flexible in our attack with him."
Abdellaoue, who joined Hannover from Valerenga in 2010, has scored 29 goals in 80 Bundesliga games for the Reds.
He believes the move to Stuttgart is a step up in his career as he joins the Europa League participants for an undisclosed fee.
"Stuttgart are a great club with many good players and a great tradition," he said on the Stuttgart website.
"This move is a big step up for me. I was delighted to receive the offer and desperately wanted to come here.
"I'm definitely delighted to be in Stuttgart. I hope that we can remain in Europe and establish ourselves in the upper regions of the Bundesliga.
"I think this has got to be the aim for such a big club as Stuttgart and I hope to have some good years here."
Stuttgart qualified for the Europa League for the second season running by reaching the final of the DFB-Pokal, which they lost 3-2 to treble-winners Bayern Munich.