They agreed to take control of the two-time European Cup winners after reaching terms to buy the late Nigel Doughty's controlling stake.
"It's an honour and privilege to assume control of this great club," said the Kuwaiti-based family, who plan to hold a news conference on Saturday.
From 1865 to 1982, Nottingham Forest were run as a members club, and were the last Football League club to switch to becoming a limited company.
"You can be assured of our best efforts in bringing the Reds back to the top."
Doughty put Forest up for sale in October but died four months later.
The Al-Hasawi's interest in the club first emerged at the end of May.
Fawaz Al-Hasawi had been chairman of Al-Quadisya Sports Club in his native country, but his decision to step down was a precursor to interest in investment within English football.
"We know there are challenging times ahead of us to bring the club back to its glory days and we look forward with excitement towards a new successful future," added the Al-Hasawi statement.
"Over the next few weeks we will be working closely with our players, on-field and off-field team members in preparation for the forthcoming season."
Doughty, a genuine Forest supporter, helped rebuild the club financially, although recovering past glories has proved difficult.
The former European Cup winners were relegated to League One in 2005. They returned to the Championship after a three-year absence but, after two failed play-off campaigns, they finished 19th last season.

The incident happened on the A92 north of Inverbervie at 11:00 on Saturday.
The 23-year-old driver died at the scene. The 13-month-old girl suffered minor injuries and was airlifted to Aberdeen Royal Infirmary.
Police Scotland said no other vehicles were involved. The road was closed for several hours but has now re-opened.

30 November 2016 Last updated at 08:06 GMT
One that has been making news because it's been hard to get hold of is the new 'Hatchimal'.
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But is it any good? We asked some Newsrounders to review it. Check out the video to hear their verdict!

Jason Downie, 32, befriended youngsters on the social media site before sending them messages asking to meet up.
Aberdeen Sheriff Court heard how he sexually assaulted a 13-year-old girl in Fraserburgh in September 2012.
Sentencing him, Sheriff Annella Cowan described Downie as a "dangerous and manipulative" man who would remain on the sex offenders register for life.
Downie, who admitted the assault, had started chatting with the young girl on Facebook and X-box Live Chat.
The court heard he then encouraged the girl to meet him in a car in Fraserburgh.
Fiscal depute Kelly Mitchell said: "The complainer asked if he would get her a sweet from the nearby shop. He said that he would do it in return for a kiss."
After seizing the youngster's head and kissing her he jumped into the back seat of the vehicle and sexually assaulted the schoolgirl.
Miss Mitchell said: "She screamed 'stop' and 'get off me'. Neighbours stated that they heard a girl screaming."
The court heard that the next day the girl went to the police and reported the attack.
Downie also pleaded guilty  to travelling "with the intention of having unlawful sex" with another young girl he befriended on Facebook earlier this year.
The court heard he sent her inappropriate messages in June 2015, asking for pictures of herself in her underwear.
At one point, he suggested they go for a drive in his car and they could kiss - but the girl turned up to the meeting near Banff in the company of friends.
The court heard Downie later sent her abusive messages, stating that she had wasted his "time and fuel".
The  girl's mother found the messages on her daughter's Facebook app and reported Downie to the police.
Downie, whose address on court papers was given as HMP Grampian, was sentenced to 38 months in jail, with an extended sentence of three years.
The Al-Hasawi family have completed their takeover of Championship side Nottingham Forest.
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A woman has died after a car with a baby on board crashed and overturned in Aberdeenshire.
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Every year Christmas comes around and with it a whole bunch of new toys.
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A man who used Facebook to find young girls to have unlawful sex with has been jailed for more than three years.