The Spaniard was introduced off the bench and made an immediate impact, sending a diving header into the bottom corner before flicking the ball on to David Luiz to set up a thrilling finale.
It marked the third time Morata has scored on his debut, netting in his first matches in both La Liga, Serie A and now the Premier League.
However, it also ended one of the longest winning runs in European football.
Incredibly, prior to taking on Burnley, Morata had been on a 52-game unbeaten streak in league matches, not having tasted defeat since a 1-0 loss to Sassuolo back in October 2015 while at Juventus.
Although Real lost three league games last season, Morata was an unused substitute on each of those occasions.
Burnley only won one home match all season last year, and have already matched that record after just one game of the 2017/18 campaign.
On their last visit to Stamford Bridge, Sean Dyche’s men had succumbed to a 3-0 loss.
Chelsea were reduced to 10 men early on in Saturday’s contest, with Gary Cahill shown a red card for a reckless lunge, and went down to nine men when Cesc Fabregas was shown a second yellow.