Westpac Banking Corporation, Australia's second-largest bank by assets, expects the transaction to complete by the end of this year.
The sale includes asset finance division Capital Finance Australia as well as corporate lender BOS International (Australia).
The motor vehicle finance book being sold stands at A$3.9 billion (£2.3bn), with equipment finance at A$2.9bn (£1.7bn) and corporate loans at about A$1.6bn (£0.9bn).
Westpac described the corporate book as "a high- quality investment-grade portfolio with exposure across Australian corporates, financial institutions and infrastructure and energy projects."
It is believed to cover more than 62,000 small and medium enterprise customers.
Lloyds, headed by Antonio Horta-Osorio, said the agreement was a further step forward in its strategy of focusing on the UK.
The bank, still 33% owned by the British government, expects to be operating in fewer than 10 countries by the end of next year, which is down from about 30 in 2011.
It added: "The sale will enable our country exit from Australia, which will be effected a short time after completion, although we will continue to support core UK-linked clients in Australia.
"The proceeds of the sale will be used for general corporate purposes."
The bank's Australian arm contributed £80m of pre-tax profit in 2012.
Lloyds said that as a result of the deal it was writing down a deferred tax asset by about £350m. However, its core capital strength is expected to improve.
The Australian arm was inherited by Lloyds as part of its rescue deal for HBOS in 2008.
Investec analyst Ian Gordon said: "Australia was probably the worse example of 'adverse asset selection' in the entire ill-fated HBOS International debacle, generating cumulative losses of over £3bn despite Australia's generally resilient performance.
Numis analyst Mike Trippitt suggested the sale would add to the prospects of Lloyds starting to pay dividends again next year for the first time since its bailout.
Westpac, which has its headquarters in Sydney and operates the largest branch network in Australia, hailed the acquisition as its largest since the A$12.6bn (£7.5bn) takeover of St George Bank in 2008.
Westpac chief executive Gail Kelly said: "This is a value-creating, straightforward transaction that makes both commercial and strategic sense.
"These are strongly performing businesses that we know well and that will expand our reach and capability in target segments."
According to Westpac, the deal will add up to A$100m (£59m) in additional cash earnings by the 2015 financial year but shareholders will see an increase in earnings per share in 2014. The purchase price is being funded from Westpac's internal resources.
Mrs Kelly added: "Our strong capital position has allowed us to expand our business without having to raise additional equity."
About 800 staff will transfer from Lloyds to Westpac, although there may be some job losses as the Australian bank is targeting synergies of A$70m (£42m).
Westpac is thought to have beaten rivals including Macquarie Bank and a consortium of Pepper Australia and GE Capital to secure the Lloyds deal.
The Australian Competition & Consumer Commission is expected to look at the deal as part of an informal merger review process, but analysts believe the likelihood of the regulator intervening is extremely slim.