Kevin Davies slotted a controversial opener when referee Howard Webb allowed Klasnic to take a quick free-kick after penalising Stoke for a back pass.
Eagles made it 2-0 with a superb strike after keeper Asmir Begovic's mis-kick.
Klasnic's 18-yard shot brought Bolton's third, a dinked Eagles finish made it 4-0 and Klasnic headed a fifth.
The result was sweet revenge for Bolton, who have endured a miserable run of 13 defeats in 16 league games since being humiliated 5-0 by the Potters in last season's FA Cup semi-final.
But as richly deserved, classy and emphatic as Bolton's first home league victory of the season was, Stoke will point to the opening goal as the beginning of a miserable afternoon.
After Begovic was pulled up for picking up a Glenn Whelan back pass, Klasnic stole the ball from the City goalkeeper's hands and took the free-kick, allowing Davies the simplest of tasks to side-foot the ball home.
Potters boss Tony Pulis was furious, seemingly both with the original award of the free-kick and the fact referee Webb let Wanderers take it so swiftly.
But his afternoon was to get much worse, and his fury was soon directed at his own team for an abject display.
A strangely subdued Stoke side, who sorely missed the energy and composure of injured winger Matthew Etherington, were desperate to avoid a fourth consecutive league loss following a midweek Europa League match.
Peter Crouch could and should have levelled soon after the hosts went ahead but he scuffed his shot and the visitors - a few dangerous set-pieces and goalmouth scrambles aside - barely threatened after that.Stoke win catalyst for Bolton's season - Coyle
Avoiding defeat never looked likely from the moment the excellent Eagles made it 2-0 when he controlled Begovic's blundered clearance and fizzed a brilliant low strike past the keeper and into the bottom corner.
Eagles and fellow winger Mark Davies both turned in majestic displays, Eagles in particular giving right-back Robert Huth a torrid time.
The two combined brilliantly to set up the third goal, laying on the ball for Klasnic, who found the bottom corner with a fierce first-time strike via the fingertips of Begovic.
By now Bolton were rampant and Fabrice Muamba and Nigel Reo-Coker both had chances to add to the home goal tally
The imposing figure of Kevin Davies then played his part in the fourth when he found striker partner Klasnic and Eagles ran on to the through-ball and cheekily lifted the ball past a bemused Begovic.
There was a still time for a fifth, Klasnic diverting in David Wheater's header following a corner.Continue reading the main story
Stoke have not claimed maximum points away to Bolton since a 1-0 victory at Burnden Park on 6 October 1990 in Division Three
And Joe Riley came close to capping a fine Premier League debut with a goal when his driven attempt flew across the six-yard box and to safety.
Bolton manager Owen Coyle:
"The performance was one of high intensity from start to finish. It was outstanding, a great performance. I shouted for the back pass. We used a bit of football intelligence, very inventive and very clever thinking. It was a clever piece of play.
"Every player, to a man, showed a level of performance today that can be a catalyst for us moving forward.
"We had to stand up and be counted today. The talking had to stop and we had to deliver. And we did. The players were up for the challenge."
Stoke City boss Tony Pulis:
"Forget the back pass. I've managed this club for over 200 games, and we've not competed today. We were second best in every department. Forget the referee, we weren't good enough. It's in the bottom three or four games in over eight years at the club.Pulis makes no excuses for 'poor' Stoke
"The way we've played is just not good enough. The players need a rest. When they come back there's certainly a few things to sort out.
"We've played Monday, then we've flown six or seven hours to Israel and back for Thursday. But that's no excuse. I'm not concerned about the first goal. I'm more concerned about not being part of the game.
"I've managed this club for over eight-and-a-half years and that's as disappointing a performance as any I've managed. We were just very, very poor all over the pitch. It's desperately disappointing, that I'm managing a team that's put in a performance like that. We were completely outfought and outplayed by Bolton, and we deserved nothing."