The supermarket's bid is a combined shares and cash offer, of 0.321 new Sainsbury's shares per HRG share and 55p a share in cash.
In addition, shareholders will receive a special dividend of 27.8p per HRG share, of which 25p will be taken from the £200m return to shareholders from the Homebase sale, and a further 2.8p will be given in lieu of a final dividend.
Sainsbury's said the offer, for a total of 173.2p per share, "represents a compelling opportunity to accelerate Sainsbury's existing strategy, with numerous expected strategic and financial benefits" including creating a "leading food and non-food retailer".
Sainsbury's chairman David Tyler said: "The UK grocery retail industry is undergoing a period of intense change in customer shopping behaviour and in the competitive environment. Against this backdrop, Sainsbury's has performed resiliently by offering great quality products at fair prices, by providing a differentiated service, and by developing strong multi-channel capabilities. All of this continues to be underpinned by our core values.
"This combination with HRG presents an opportunity to accelerate our strategy, delivering compelling revenue and cost synergies. We will create a multi-product, multi-channel proposition with fast delivery networks that we believe will be very attractive to the customers of both businesses."
The final bid comes just an hour after Steinhoff, which is backed by South African tycoon Christo Wiese, revealed it would not be making a formal offer.
In a statement to the markets, Steinhoff said: "For the purposes of note 2 on rule 2.8 of the code and in the event that Sainsbury's announces a firm intention to make an offer for HRG during the current offer period, this announcement will be treated as having been made after the Sainsbury's firm offer announcement.
"The UK panel on takeovers and mergers has agreed that, notwithstanding the provisions of panel statement 2016/4, Steinhoff is no longer to be treated by it as a potential competing offeror for HRG for the purposes of the code and no further clarification announcement by Steinhoff will be required."
Here's what the various announcements have been doing to Sainsbury's and Home Retail Group's share price
18 March 2016 @ 4:30pmHome Retail Group (HOME)
18 March 2016 @ 4:30pmSainsbury (J) (SBRY)
Sainsbury's has made no secret of its desire to secure the business, offering £1.3bn for it back at the start of February.
Just yesterday the Takeover Panel said another offer made before the 5pm deadline would result in the deadline being automatically extended yet again.
Instead of Home Retail Group, Steinhoff has confirmed its bid for Darty, which it is acquiring for 125p per share in cash, valuing the business at £673m.
Shareholders will receive a 54.3 per cent premium to the price of a Darty share held at 29 September, when Steinhoff first revealed it was eyeing the firm.
Darty's share price initially plunged on the news, but recovered almost immediately and is now trading up one per cent.