Victory maintained the Baggies' 100 percent start to the season but for Burnley, relegated from the Premier League last term, it was a second game without a win following a 2-2 draw with West Ham last week.
Northern Ireland international Price scored twice in the second half after setting up Harry Whitwell's equaliser to cancel out Callum Style's own goal.
West Brom staved off relegation from the Championship in their last campaign.
"We've come into this season fresh, we've got some new young players who look hungry and it's working for us at the moment," Price told Sky Sports.
Baggies boss James Morrison added: "I'm not one for giving credit, but credit to the lads, it was a top performance.
"Just with our intensity and attacking play but also the defending. Lads running back, duels, on the sideline I enjoyed watching it."
Styles put into his own net in the eighth minute from Jacob Bruun Larsen's cross as visitors Burnley went ahead.
Whitwell, however, drew West Brom level 10 minutes later when he struck with a low shot from 12 yards out as the 20-year-old scored his first league goal for the club.
West Brom dominated the second half and they were ahead just a minute after the interval when Price curled a shot beyond Max Weiss, after the Clarets goalkeeper had strayed off his line, and into the far corner.
Price sealed victory from the penalty spot in the 65th minute after Bashir Humphreys upended Whitwell.
"First of all it (the defeat) was totally deserved," said Burnley manager Nicky Hayen. "They had much more energy than us."
He added: "We played too slow, we were far away from the level last week. We said it in the dressing room - if we play at our best we're really a top team, but if you play like this then you're just an average team."