The Giallorossi occasionally made it tough for themselves - especially considering Lecce were reduced to 10 men early on - but ended up getting the all-importan three points, as Dybala converted the winner from the penalty spot after Tammy Abraham had been fouled.
It was Smalling who opened the scoring early in the first half, planting a towering header home inside the first six minutes.
Lecce's response was limited but the one real chance they had they took; as Gabriel Strefezza lashed home from inside the box just before the interval.
Jose Mourinho's side came out for the second half in determined form, however, as substitute Tammy Abraham immediately made his presence known as he was fouled inside the box.
Dybala duly converted from 12 yards, only to seemingly hurt his left thigh in the process, forcing his substitution.
Nevertheless, Roma duly saw out the rest of the contest to continue the momentum created by last weekend's win over Inter Milan.
The side are back in action on Thursday: they visit Seville to take on Real Betis in the Europa League.
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