Going into the game 1-0 up from last Thursday's initial meeting in Switzerland, the Potters soon extended their lead, scoring three times in the first half through headers from Matthew Upson and Jones and a drilled effort by Glenn Whelan.
Jones crashed home his second with just under 20 minutes to go before substitute Andreas Wittwer netted a consolation for Thun.
The result means Stoke, who are taking part in major European competition for the first time in 37 years, have guaranteed themselves at least six more continental fixtures and they will discover who their opponents are when the draw for the group stage is made tomorrow.
Manager Tony Pulis made four changes to his starting line-up, replacing suspended duo Robert Huth and Matthew Etherington with Andy Wilkinson and Danny Pugh, scorer of the only goal in the first leg.
The other adjustments from Sunday's Barclays Premier League draw at Norwich saw Thomas Sorensen and Upson included in the first XI again as they had been a week earlier, while Jermaine Pennant was fit to start despite limping off in the first half at Carrow Road.
Thun exerted some early pressure and Marc Schneider's shot was blocked by Wilkinson in the third minute.
Stoke began to settle down but struggled to fashion much going forward, with Pennant seeing a delivery into the danger zone headed clear by Benjamin Luthi and Dean Whitehead just failing to get on the end of a return pass from Jon Walters in the box.
Muhamed Demiri tried a spectacular effort from distance for the visitors which sailed high over the bar before the Potters carved out their first real goalscoring opportunity, Jones nodding Pennant's cross wide in the 23rd minute.
Two minutes later Stoke were in front as Pennant swung in a corner and Upson rose to head in his first goal for the club on his home debut.
All the Swiss side could muster in response was another ambitious long-range strike, launched by Stephan Andrist and easily caught by Sorensen, before Pulis' men further increased their advantage just after the half-hour mark.
Pennant was the supplier again, clipping the ball into the area where a diving Jones directed it under stand-in Thun goalkeeper Dragan Djukic.
Djukic - in the team because regular stopper David Da Costa was banned following his sending-off in the first leg - was then beaten a third time, a failed clearance from a corner in the 38th minute falling to Whelan, who lashed into the net from the edge of the box.
Four minutes after the restart Wilkinson battled his way past several players to unleash a shot that flashed wide.
Thun captain Roland Battig hit a tame effort which Sorensen claimed comfortably.
Pennant was then taken off for Tom Soares, and Pulis also subsequently introduced Danny Collins and Ryan Shotton in place of Marc Wilson and Walters.
The addition of fresh legs paid off in the 72nd minute as Soares beat his man on the right and sent in a cross which was converted by the head of Jones.
The Potters appeared to ease off slightly in the closing stages and Wittwer managed to pull one back for the visitors with 13 minutes of normal time to go.
Sekou Sanogo was then allowed to head wide, but Stoke's job was done, the final whistle confirming the continuation of their European adventure.