Britton has been a key component of the club's rise to the Premier League but has also heaped praise on his former boss
Jackett was sacked by Wolverhampton Wanderers on Friday night, denying him a reunion with his former club at Molineux this weekend, where the two teams will meet in a pre-season friendly.
The Welshman's future in the Black Country had been the subject of speculation since new owners Fosun International completed a deal for the club, with his departure eventually confirmed via an announcement on their official website.
But Britton, who played under Jackett in South Wales, has come out in support of his former boss, claiming he deserves credit for playing his part in helping Swansea on their impressive rise from the fourth tier of English football to the top flight.
He told the club's official website: “Kenny played a big part for us, and maybe he does get missed out a little bit when people talk about our climb up the leagues.
“People talk about Roberto (Martinez) and rightly so, but Kenny got us out of League Two and he very nearly got us out of League One as well. We were just a few spot-kicks away. Things could have worked out so differently if we had won that shootout."
Jackett took charge of the Swans in 2004 and enjoyed a successful three-year spell at the club, leading them out of the League Two during the final season at Vetch Field, before missing out on promotion to the Championship due to a penalty shootout defeat to Barnsley in 2006.
He departed the club in 2007, but was a big factor in helping bring Britton into the first team fold as part of a new approach that the midfielder believes had a big say in their success.
“Under Brian Flynn we were very much an attacking team,” he added.
“When Kenny came in, it was a more rigid 4-4-2. We were more drilled and organised, and I think that showed in the year we went up.
“Kenny definitely played a big role here. He may go under the radar a bit, but everyone who has been at the club for a long time or supports the club appreciates what he did for us.”
Attention at Wolves will now turn to appointing a successor, with ex-Olympiakos boss Marco Silva emerging as the early favourite.