The Clarets had made their worst-ever start to a top-flight campaign, picking up one point from the first 15 on offer, but an improved display at Turf Moor helps lift them up to 16th in the table.
It may have been a different game altogether had Nathan Ake's shot found the back of the net inside the opening quarter of the match, with the ball instead deflecting off Ben Mee and onto the crossbar.
Burnley, aiming to avoid a sixth defeat on the spin, wasted a glorious chance of their own soon after as Sam Vokes failed to convert at the back post when the ball got stuck under his feet.
Two quickfire goals before half time gave Burnley some much-needed momentum, though, with Vydra smashing the ball into the net from inside the six-yard box after Ashley Westwood's shots were blocked.
Johan Berg Gudmundsson then delivered a teasing ball towards the back post for a lurking Lennon to slide in and double his side's tally, putting the Clarets on course for a first league win in 11 attempts.
Comeback kings Bournemouth had collected 25 points from losing positions ahead of this game, but they were unable to find a route back into the contest at Turf Moor.
Vokes sent a half-volley wide of the target as the hosts looked to punish some poor defending from their opponents, but the Cherries did manage to grab more of a foothold in the game as it wore on.
Some good goalkeeping from Joe Hart prevented David Brooks from pulling one back on the hour, racing out and blocking the ball to thwart the Welshman, who had Joshua King alongside him had he looked across.
Despite turning the screw, though, Bournemouth struggled when it came to finding a way through on goal and the game was soon put out of their reach.
Lennon skipped past his man and picked out Gudmundsson, whose shot hit the post and fell invitingly for Barnes to convert from close range.
Substitute Barnes added his second late on, this time stroking the ball out of Asmir Begovic's reach following some more good play from Lennon out wide on the counter.
BURNLEY (4-4-2): Hart; Lowton, Tarkowski, Mee, Taylor; Lennon, Westwood, Cork, Gudmundsson; Vydra (Wood 60'), Vokes (Barnes 68')
BOURNEMOUTH (4-4-2): Begovic; Smith (Defoe 78'), Steve Cook, Ake, Rico (Francis 46'); Brooks, Surman, Lerma, Fraser; King, Wilson (Stanislas 68')