The result marks the end of Burnley's three-game unbeaten run as they drop one place to 12th in the standings after as many games in the top flight.
It was the perfect start for West Brom as Phillips put the team ahead in the fourth minute after turning in the box and prodding the ball home from close range.
Morrison was the man to cross it into the danger area, and even though Dean Marney got a touch on it, he could only help it along to Phillips.
The visitors had plenty of possession - a rare situation for Sean Dyche's side - but they struggled to get through a packed Baggies half and they were very vulnerable on the counter-attack, which is how the second goal was produced.
Brunt led the way as he played the ball forward for Rondon, who used his strength to hold up the ball before setting up Morrison. The midfielder completed the move by charging forward and drilling a shot into the bottom corner.
Conceding twice in 16 minutes stunned Burnley and even the keeper seemed to suffer from a few nerves as he ended up giving the ball straight to Morrison while under the pressure of Rondon, but thankfully for the shot-stopper, the midfielder's shot looped over the crossbar.
Heaton recovered from his blunder and would have gained some confidence after denying a goal from Rondon, who let one fly from the edge of the box, but the Burnley keeper leaped across to palm the ball away from the top corner.
West Brom's impressive first half was capped off with a goal from Fletcher, who met Rondon's cross at the far post and fired it into the roof of the net to give his team a three-goal advantage.
It was a disappointing goal to concede from Burnley's point of view as they allowed the team's defence to be stretched, while also failing to put a man on Fletcher, who was completely unmarked when he hammered it home.
After the break, West Brom had an early opportunity when a high cross found Phillips in plenty of space in the box, but his looping header dropped behind the goal.
Dyche changed Burnley's shape by withdrawing Steven Defour and bringing on striker Ashley Barnes, whose early involvement resulted in the team's first corner of the game after he turned in the box before forcing Claudio Yacob to clear.
The resulting set piece came to nothing, though, as West Brom comfortably cleared and continued to cause problems to their opponents on the break.
After an influential 64 minutes, Rondon was rewarded with a goal of his own as Morrison's first-time pass found Allan Nyom out wide, and the defender edged to the byline before cutting it back for the attacker.
Rondon showed composure as he swivelled before drilling it home, and even though it was helped along by a couple of deflections, the Venezuelan had no complaints.
A rare opportunity fell to Burnley in the final 15 minutes when the ball dropped to Johan Berg Gudmundsson on the edge of the area, but the midfielder's stinging shot was directed straight at Foster.
The Iceland international had another chance a few minutes later as he forced Foster into tipping the ball over the bar after delivering a free kick from just outside of the area.
Goalscoring chances were few and far between for the visitors, who have lost half of their league games so far this season.
In the end, West Brom comfortably saw out the win and could have increased their lead as they peppered the Clarets' goal until the final whistle.