Kane netted from the spot in the first half after Glenn Murray was adjudged to have handled the ball before substitute Lamela latched onto a cross to double the lead in the second half.
As always, Sportsmail is on hand to point out some of the things you may have missed on the south coast.
Eric Dier’s catwalk
Mauricio Pochettino’s Tottenham side were kitted out to the nines on Saturday afternoon when they arrived at the AMEX Stadium.
Donning black from head to toe, the Spurs boys sauntered into the stadium in the sharpest of tailored suits.
Midfielder Eric Dier, however, got caught up in it all and appeared to be modelling the outfit in a catwalk out of the tunnel, which was captured in an extremely candid snap.
Eric Dier appeared to be modelling his suit in a catwalk out of the tunnel at the AMEX Stadium
He then proceeded to take selfies with adoring fans who were calling his name in the stands
True Seagulls fan
Donning your club’s colours or wearing a scarf with the crest emblazoned upon it is commonplace at football matches. One Brighton supporter, however, took his loyalty to the next level on the south coast.
A fan was pictured behind the Brighton goal in the first half — while Harry Kane was preparing to lash home his penalty — fully masked with a seagull’s head.
One Brighton fan donned a seagull’s head as a mask to show their true support at the AMEX
Spurs pay tribute to Chas
The awful news that Chas Hodges, of Seventies musical duo Chas and Dave, had died filtered through on Saturday afternoon after his battle with oesophageal cancer.
Chas was a big Tottenham fan and the club even paid tribute to him ahead of the game.
As a touch of class by all Spurs supporters at the AMEX, applause rang out on the south coast to honour one of London’s most iconic singers.
Tottenham paid tribute to Chas Hodges after his passing before fans applauded at the AMEX
Chris Kavanagh fails at Dele Alli celebration
Referee Chris Kavanagh was rather animated when he awarded Tottenham the first-half penalty, raising his hand to show clearly he though Glenn Murray handled the ball.
Some eagle-eyed fans, however, believe Kavanagh was not simply telling Brighton the reason for which he awarded the penalty – they think he was attempting Dele Alli’s now famous celebration! Not quite mastered it yet, Chris.
Some Twitter users pointed out that Chris Kavanagh failed in his bid to emulate Dele Alli
Alli’s celebration has now become famous this year and began a trend throughout the country
Kane joins the 143 club
Harry Kane has now moved up to joint-fifth in the all-time goal-scoring tally for Tottenham. His penalty against Brighton means he has now netted 143 times in a Spurs shirt, moving him level with Jermain Defoe.
Cliff Jones (159), Martin Chivers (174), Bobby Smith (208) and Jimmy Greaves (266) are still ahead of him in the list.
Harry Kane’s goal saw him move up to joint fifth in the all-time goalscoring list for Spurs
Blood, sweat and tears for Dunk
Lewis Dunk copped an accidental elbow in the face from Erik Lamela late on in the game on Saturday evening.
The Brighton defender hit the deck before blood began to pour out of his nose. He appeared to be alright after the incident – but that will certainly hurt in the morning.
Lewis Dunk copped an elbow to the face late on against Spurs on Saturday evening
The Brighton defender hit the deck before blood began to pour out of his nose