Sean Dyche's side went into the game on the back of four straight top-flight defeats and looked every bit the league's bottom side for much of the opening period.
It burst into life shortly before the interval, though, as Vydra and Lennon opened their accounts for the club with close-range finishes within the space of two minutes.
Joe Hart made a fine stop from David Brooks to deny an out-of-sorts Bournemouth the opportunity of a grandstand finish before two Barnes strikes in the final seven minutes gave the scoreline an emphatic look.
The host had started nervously and was lucky not to go behind in the 17th minute when Nathan Ake's heavily deflected effort looped onto Hart's crossbar.
Burnley had offered little in the way of attacking threat for most of the half, but two quick-fire goals before the break gave them a commanding lead.
Vydra marked his first start for the club following his move from Derby County with the opener after 39 minutes, slotting home from close range after Ashley Westwood had seen two shots blocked.
That was followed just two minutes later by Lennon's strike. The former Everton winger stole in at the back post to meet Johann Gudmundsson's teasing cross with a controlled half-volley that nestled in Asmir Begovic's bottom-left corner.
Any thoughts Bournemouth had of mounting a late comeback were extinguished in the 83rd minute when substitute Barnes reacted quickest to Gundmundsson's strike which had cannoned back off the post.
He followed that up five minutes later with a cool finish from Lennon's pull-back to complete a superb afternoon for Dyche's men.
What it means: Burnley up and running
Before the game Dyche had spoken about the need to lift the fog surrounding the club. Well, the Burnley boss can look forward to a few days of clear skies after an impressive win over the in-form Cherries. Eddie Howe, meanwhile, will be left scratching his head over a performance and result that takes some shine off their impressive start to the campaign.
Lennon instrumental for Clarets
After a difficult few years, Lennon looked back to his very best with an effervescent display. He showed great composure for his goal and played vital roles in his side's third and fourth strikes to complete a memorable afternoon.
Begovic provides little resistance
The Cherries goalkeeper did not make any howlers for Burnley's goals, but it is never a good look when you concede four goals from just five shots on target. His uncertainty in coming for Gudmundsson's cross definitely made Lennon's finish a lot easier.
Both sides are in EFL Cup action on Tuesday as Burnley travels to Burton Albion and Bournemouth hosts Blackburn Rovers. In their next Premier League games, Dyche takes his side to Cardiff City next Sunday, while the Cherries welcomes Crystal Palace to the Vitality Stadium the following day.
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