The 34-year-old - the most capped keeper in Wales history, making 96 appearances - arrives as a free agent after leaving Crystal Palace, and has signed a two-year deal – with a further year’s option.
Hennessey has made 181 Premier League appearances for Palace and Wolves, where he was also a Championship title winner in 2008/09 early in his career.
After more than a decade with the West Midlands club – which included playing loan spells with Stockport County, where he set a then-Football League record of nine successive clean sheets, and Yeovil Town – the six-feet, six-inch keeper then joined Palace in January, 2014.
Hennessey – who becomes the Clarets’ second senior summer signing after defender Nathan Collins – went on to become first-choice keeper for the Eagles as they went about securing an uninterrupted eight-year run in the Premier League.
More recently he provided competition and cover for Vicente Guaita at Selhurst Park and took his number of appearances for the Londoners to 132 before ending his seven-and-a-half year stay with Palace last month.
The signing is likely to mean Bailey Peacock-Farrell being allowed out on loan to get some game time, with Huddersfield Town and Sheffield Wednesday among the interested clubs.