Greenwich-born Salter made nearly 50 appearances for the West Country club in his four seasons at the Memorial Ground.
He has previously played with West Hartlepool and Blackheath in Rugby Union and also spent three seasons in Rugby League's Super League for London Broncos, including playing against the Tykes' sister club Leeds Rhinos in the 1999 Challenge Cup Final at Wembley.
The 27-year-old Salter follows former Bristol team-mates Phil Christophers, Scott Morgan and David Rees – becoming the fourth player from Bristol to make the move to Headingley during the off-season.
Tykes coach Phil Davies was happy with the addition of Salter to the squad and felt the big flanker would play an important part in the season.
"I am delighted Matt has agreed to join us and I know he is looking forward to the challenge of playing for Leeds Tykes," he said.
"We have a very competitive back row now including Dan Hyde, who is currently away with England 'A', Alix Popham, who is part of the Welsh World Cup squad and James Ponton. Add to that our new signings in Aaron Persico and Colm Rigney and this will certainly be an extremely important part of our side for next season.
"Matt is a very strong player who has pace and mobility around the field. He has a tremendous work-rate and is an accomplished tackler, who has developed his style whilst playing Rugby League earlier in his career."
Davies has just returned from a coaching seminar in New Zealand and was looking forward to employing some of what he had learnt.
"It was a great experience to go out to New Zealand and spend some time with some of the best coaches in the world. I have picked up a lot of ideas which we can hopefully incorporate into our operation.
"The players still have another couple of weeks off before pre-season starts and then we will be building up for what could be the biggest season ever in the club's history."