Lock Jones, who has 13 Welsh caps to his name, has missed recent games due to a shoulder injury allowing Connacht's most-capped player Swift to claim the top spot.
'Swifty' joined Connacht in 2000, making his debut against Ulster while playing alongside current Connacht coaches Eric Elwood and Dan McFarland.
He is one of an elite group of players that were part of the very first Celtic League in 2001.
Speaking about his achievement of playing 140 times in the league, he said: "I'm obviously over the moon to reach this milestone. The league has gone from strength to strength over the years as has the quality of the rugby played, so as a brand the RaboDirect PRO12 stands in a great place at the moment."
When pressed about what keeps a rugby player going for so long in the physically-demanding modern game, Swift added: "I suppose you could put my longevity down to a mix of good fortune and a great medical team looking after me."
Known for his ferocious tackles and towering lineout presence, the colossal 6ft 5in, 120kg lock shows no sign of slowing down just yet.
He hopes to add to his already impressive tally of caps when Connacht take on Munster at the Sportsground in a mouth-watering interprovincial derby on Saturday, March 24.