Ronaldo denied his Portuguese boyhood club a famous win by scoring an equaliser from a free-kick in the 89th minute just when it seemed Madrid could, sensationally, lose their opening game in Group F.
Morata then headed home deep in injury time to complete the fightback.
"We didn't start the game well but Real Madrid are famous for comebacks like these and we leave here very happy," said Morata.
"We always believed in ourselves and we proved that you can always win a game until the referee blows the final whistle. Football is not always fair but it was fundamental to win and that's what we've done."
Brazilian Bruno Cesar had earlier given the visitors a surprise yet deserved lead in the 47th minute as he pounced on indecision in the Real back line.
Real took their time to respond to going behind with Ronaldo missing a gilt-edged chance to level when he struck the near post from a couple of yards out.
Real had gone into the game on top of La Liga after a convincing 5-2 win over Osasuna but met a well-drilled Sporting side who flooded the midfield with bodies and frequently won possession back, launching quick and dangerous breaks.
The first of these breakaways came early in the game and led to Cesar trying his luck from the edge of the area, firing narrowly wide of Kiko Casilla's right-hand post.
Real struggled to find solutions to Sporting's strategy and were limited to shots from distance, like one stinging effort from Ronaldo which Rui Patricio tipped over the bar.
Gareth Bale had looked Real's liveliest player in the first half but was on the wrong end of a painful challenge from Sebastian Coates, who kneed the Wales international in the midrift.
Bale was taken off in the second period along with Karim Benzema, who looked far from peak fitness.
Zidane's double substitution had a tangible impact on the game, however, with Lucas Vazquez giving Real extra energy while Morata added extra power in the area.
And it was the local striker's tenacity that won the points at the death as he leapt to head James Rodriguez's cross home.
(Reporting by Richard Martin; Editing by Ian Chadband)
