Hendrick struck from long range in the 13th minute and further goals from Stephen Ward and George Boyd secured the three points despite Benik Afobe and Charlie Daniels replying for Bournemouth.
Victory ended a run of three consecutive defeats for Sean Dyche's Burnley, who are now five points above the relegation zone and a point below Bournemouth, who slipped one place to 11th.
Bournemouth had stormed back from 3-1 down to beat Liverpool 4-3 on their previous outing, but manager Eddie Howe could not engineer a repeat performance against his former club.
Ryan Fraser was rewarded for inspiring Bournemouth's comeback against Liverpool with a starting place as the south-coast club went in search of a first ever win at Turf Moor at the 14th attempt.
His early run created a chance for Afobe, who was brilliantly denied by fit-again Burnley goalkeeper Tom Heaton.
Dan Gosling also put a shot over Heaton's crossbar, but it was Burnley who drew first blood, and in stupendous fashion.
Racing down the inside-right channel, Hendrick leapt to bring down a pass from a team-mate, kneed the ball up and then cracked an unstoppable, looping volley into the top-right corner from 25 yards.
His fellow Irishman Ward added a second less than three minutes later, slamming home after Bournemouth goalkeeper Artur Boruc failed to hold Ben Mee's header.
Boruc then had to save from Michael Keane, but after Adam Smith had extended Heaton with a free-kick at the other end, Afobe tucked away Simon Francis's cut-back to reduce the arrears in first-half injury time.
After Afobe had squandered a good chance to make it 3-1, scuffing hopelessly wide, Dyche turned to his bench, sending on Andre Gray and Ashley Barnes for Sam Vokes and Steven Defour.
Gray procured a clear opportunity to stretch the hosts' lead when the ball fell to him in a kind position, but he thrashed his shot straight at Boruc.
He made a more telling contribution in the 75th minute, his back-heel teeing up Boyd to double Burnley's lead, and although Daniels hit back for Bournemouth in stoppage time, it came too late.