Westpac's chairman, John Uhrig, said Mr Joss, 51, would take up his post in February.
'Mr Joss meets the exacting criteria set down by the board for the task ahead,' he added.
Mr Conroy resigned in December following what analysts said was a disagreement with new 10 per cent shareholder Kerry Packer over the extent of cost-cutting at the loss-making bank.
Mr Packer and the managing director of his Consolidated Press Holdings group, Al Dunlap, later resigned as Westpac directors after less than a week on the board.
The bank had planned to name Mr Conroy's replacement by the time of the annual meeting last Tuesday, but Mr Uhrig announced at the meeting that discussions were not complete.
Mr Uhrig said Mr Joss would bring extensive experience to Westpac through his association with Wells Fargo since 1971.
'He will bring a fresh approach and provide momentum for the strong recovery strategy already in place,' he said.
Mr Joss was appointed Wells Fargo vice-chairman in 1986. Among other duties he is in charge of international trade services, private banking, consumer credit and corporate strategy.
Mr Joss said Westpac had a strong retail franchise and key core activities in institutional and offshore banking which had great potential.
'I am looking forward to taking on this exciting and challenging task, meeting with staff and working with everyone to rebuild Westpac,' he added.
'I believe it is a bank with a tremendous future and I am determined to ensure that it regains its position as a strong and profitable bank.'
Mr Joss was considered a frontrunner for the post, along with the former Bank of New Zealand chief executive Lindsay Pyne.
Speculation about Mr Joss's appointment had already stirred up some controversy in the media about the suitability of a banker with no local experience taking on the job of reviving Westpac.
The turmoil at the 175-year-old bank follows its record loss of Adollars 1.56bn (pounds 685m) for the year to 30 September 1992.
Last week's annual meeting, attended by more than 5,000 people, was adjourned until January 27 after more than eight hours of heated discussion on the bank's performance and board reappointments.
The meeting has yet to consider the reappointment of Mr Uhrig, who took over in October when the former chairman Sir Eric Neal and four other directors quit. Nine directors have left the bank in recent months.