Spurs, who thrashed Newcastle United 4-0 in the Capital One Cup quarter-finals in midweek, took the lead after 21 minutes when Harry Kane headed home his first home league goal.
Burnley levelled five minutes later thanks to a brilliant strike from Ashley Barnes that flew into the top corner.
But Lamela restored Spurs’ lead with an even better effort after 35 minutes when he curled a wonderful shot into the far corner and out of Tom Heaton’s reach.
Spurs survived a nervy second half with their one-goal lead intact to win only their fourth home league game of the season and move above Arsenal and up to sixth in the table.
It was the perfect end to a good week for Spurs, who won at Swansea last weekend before sweeping aside Newcastle to reach the semi-finals of the League Cup.
They only made one change from their victory at the Liberty Stadium, with Nacer Chadli starting in place of Roberto Soldado.
Burnley were unchanged from the side that beat Southampton last time out and they could have taken the lead inside two minutes as Kieran Trippier picked out Scott Arfield with a back-post cross, but he failed to hit the target on the half volley.
Spurs also threatened during a lively start to the match as Kane went for goal on a couple of occasions and Ryan Mason missed the target from the edge of the box.
When the breakthrough did arrive after 21 minutes it was thanks to some quick thinking from Christian Eriksen.
The Dane was a few yards offside when a quick free-kick was played through to him so opted to leave the ball for Chadli, who beat a defender and then crossed for Kane to head home from close range.
It was Kane’s first Premier League goal at White Hart Lane but it was quickly overshadowed as Barnes levelled with a stunning strike five minutes later, collecting George Boyd’s pass and firing an unstoppable shot into the top corner.
Kane should have quickly restored Spurs’ lead but failed to make contact with the ball from Lamela’s pull back with the goal at his mercy.
Jason Shackell then headed wide at the other end when he should have hit the target before Lamela made it 2-1 with a wonderful effort.
Cutting in from the right he picked his spot in the far corner and curled a superb shot beyond the dive of Heaton.
Mason went close to making it 3-1 as he fired over the bar, but the midfielder then had to be taken off before half-time with an injury and was replaced by Benjamin Stambouli.
After the break Burnley almost equalised immediately as Danny Ings twice went close in a matter of minutes.
The England Under-21 international saw a near-post shot saved by Lloris after beating Fazio for pace and then was denied again by the keeper from the subsequent corner.
Burnley continued to look the livelier side as Spurs struggled to ease their fans’ nerves with a third goal.
It took them until the final 10 minutes to create any clear openings but even then crosses from Kyle Walker and Eriksen didn’t find a team-mate.
Fortunately for the home side their defence largely kept Ings and Barnes quiet and Lloris was only called upon to save a late free-kick from substitute Ross Wallace.
Tottenham Hotspur: Lloris (7), Walker (6), Fazio (6), Vertonghen (6), Davies (6), Bentaleb (5), Mason (5), Lamela (8), Eriksen (6), Chadli (7), Kane (7).
Subs: Rose (5), Soldado (5), Stambouli (6).
Burnley: Heaton (6), Trippier (6), Keane (6), Shackell (5), Mee (5), Boyd (6), Marney (6), Jones (6), Arfield (6), Ings (6), Barnes (7).
Subs: Wallace (5), Kightly (5), Jutkiewicz (5).
Man of the match: Erik Lamela.