Swansea made a strong start against Burnley, winless away from home this season, when Spain forward Llorente headed home in the 12th minute for his 10th goal of the season.
Andre Gray restored parity eight minutes later when he converted a controversial penalty - awarded despite a handball by Burnley forward Sam Vokes rather than Swansea's defence - before firing home his second after the break.
Swansea struck the woodwork three times before Martin Olsson equalised with a powerful drive and Llorente's late header sealed their fourth win from six league games. In-form Swansea, in 16th, have 27 points from 27 games while Burnley, in 12th, have 31.
(Reporting by Tom Hayward, editing by Ed Osmond)
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