The Argentinian, who arrived for a club-record fee from Roma prior to the 2013-14 campaign, had previously gone 25 league appearances without finding the net for Spurs, but he broke his duck in fine style on Saturday.
Lamela produced a sublime curling finish from 25 yards to net the winner in the 35th minute, after a strike of similar quality from Ashley Barnes had cancelled out Harry Kane's header.
Kane met Nacer Chadli's pinpoint delivery midway through the first half - his 14th goal of the season in all competitions and third in as many matches.
Barnes netted the winner in Burnley's 1-0 victory over Southampton last weekend and, although there was a slice of fortune about that strike, there was nothing lucky about his third Premier League goal as he hit a magnificent effort into the top corner from just outside the area.
That goal was outshone by Lamela's effort, however, as Spurs completed a successful seven days, having defeated Swansea City last weekend and progressed to the semi-finals of the League Cup with a 4-0 thrashing of Newcastle United on Wednesday.
The hosts showed one alteration from their 2-1 victory in south Wales as Chadli came in for Roberto Soldado.
Burnley, unchanged from their clash with Southampton, threatened an early opener when Scott Arfield fired off target from Kieran Trippier's deep cross in the second minute.
After that promising start for the visitors, Spurs began to assert themselves on the game and took the lead in the 21st minute.
Kane's quick thinking at a free-kick released Chadli down the left and the striker broke into the box where he met the Belgian's perfect cross with a simple header from close range.
The advantage was short-lived, however, as Barnes arced a delightful finish into the top right-hand corner from 20 yards six minutes later.
Kane ought to have restored his side's lead soon afterwards, but he somehow missed his kick inside the six-yard box after Lamela had cut the ball back from the byline, Christian Eriksen eventually blazing over.
Ten minutes before the interval, Lamela's moment of magic did ensure Spurs moved back in front as he cut inside on his left foot and curled a superb shot inside the far post beyond the despairing dive of Tom Heaton.
Ryan Mason injured his ankle late in the first half while shooting off target and had to be replaced by Benjamin Stambouli.
As in the first period, Burnley made a strong start in the second with Danny Ings denied three times by Hugo Lloris in the opening nine minutes.
Spurs continued to look a threat themselves and Heaton stood strong to deny Chadli, who had been played in by a fine Eriksen backheel.
Kane came closed to a third late on when he just failed to get a telling touch on Eriksen's low cross, before Lloris kept out substitute Ross Wallace's low free-kick in stoppage time, allowing Spurs to climb to sixth, while Burnley drop back into the bottom three on goal difference.
