The former England Under-21 goalkeeper was out of contract at Ashton Gate and his move north has been widely reported over recent days.
Heaton has agreed a two-year deal with the option of a further 12 months and is poised to be Clarets boss Sean Dyche's number one next term following the departures of Lee Grant and Brian Jensen at the end of the recently completed campaign.
Dyche said: "Tom is someone we knew all about at Watford and we have monitored his progress since then.
"We have had him watched several times in the past year, he's just had a really good season in very tough circumstances at Bristol and he recently played very well here at Turf Moor, as every Burnley fan will know.
"He is a good technician, he has great pedigree, having coming out of Manchester United and is a good age and willing to learn."