The midfielder wrapped up a 4-2 victory at the King Power Stadium which left Leicester top of the early table on goal difference.
Leicester maintained their 100 per cent record following a 3-0 win at West Brom last week – the first time they have won their opening two league games since 2015, when they went on to win a shock title.
Burnley opened their top-flight campaign with defeat and desperately missing the injured James Tarkowski.
Sean Dyche's side started well enough and threatened early when Wood was sent clear by Robbie Brady, but his lack of pace saw the chance wasted and Wilfred Ndidi stepped in to mop up.
But Wood and Burnley's positive start was rewarded when the striker opened his account after 10 minutes.
Charlie Taylor's deep cross found Wood at the far post and he outmuscled Justin to take a touch and fire into the bottom corner.
VAR gave the goal after a handball check while Leicester also felt Justin had been pushed.
But the lead lasted just 10 minutes as Barnes grabbed his first goal since March.
Pre-game Rodgers had urged him to score more to gatecrash the England squad ahead of next year's delayed European Championships and the winger immediately heeded his advice.
His initial ball to Jamie Vardy created havoc but when Burnley blocked him off the striker found Timothy Castagne to tee up Barnes to roll in.
Confident Barnes was Leicester's biggest threat and Nick Pope denied him a second with a fine low save as the hosts took control before Phil Bardsley steered Praet's cross behind with Barnes ready to pounce.
Burnley, without long-term Leicester target Tarkowski who was missing with a toe problem, struggled to contain their hosts and they lost Brady to injury before half-time.
Youri Tielemans shot straight at Pope ahead of the break but Leicester made their dominance count five minutes after the restart.
A brilliant defence-splitting ball from Tielemans picked out Castagne and the defender's low cross was inadvertently turned in by Burnley substitute Pieters.
The Clarets' chances had been limited since Woods' opener but the New Zealand international blew a fine opportunity to level after 59 minutes.
A neat move involving Josh Brownhill and Jay Rodriguez ended with Wood scuffing his shot straight at Kasper Schmeichel.
He was left to rue the miss when, two minutes later, Leicester moved further ahead.
Without Tarkowski Burnley looked brittle and Ayoze Perez exposed them when he tricked his way past some half-hearted challenges and his deflected cross found Justin who swept in his first top-flight goal.
It should have eased any nerves but Dunne – making his Premier League debut – headed in Dwight McNeil's free-kick with 17 minutes left to make it 3-2.
But Praet rifled into the top corner from the edge of the area 11 minutes from time to seal the points.
Wood still had time to hit the post but Burnley were well beaten.