Swansea, who were beaten in the Championship play-off final in May, turned to Eustace after head coach Steve Cooper left earlier this week.
The Welsh club will pay a small compensation fee to Championship rivals QPR, though Eustace is expected to remain with QPR for their pre-season friendly against Manchester United on Saturday.
The Swans face Blackburn in their first game of the new Championship on Saturday August 7.
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