The pair, who scored all five between them in last week’s Europa League defeat of Asteras Tripolis, grabbed a goal apiece to book a quarter-final showdown with Newcastle United at White Hart Lane.
Lamela came off the bench to put the finishing touch to a superbly worked goal on 54 minutes, while Kane showed his goalscoring instincts to pounce on a loose ball 20 minutes later in what was a relatively comfortable victory.
We were looking for an immediate response after Sunday’s disappointing defeat to Newcastle, while Albion arrived at the Lane on the back of a 10-game winless run in the Championship.
We were very much on top in the early stages, even if clear-cut opportunities were not flowing freely, with the visitors defending well, throwing bodies in the way on a number of occasions.
Kane saw a third-minute drive deflected for a corner, while Aaron Lennon and Andros Townsend combined well inside the area, but the latter’s cutback from the byline was blocked.
Kyle Naughton’s fizzing shot cannoned off a defender for another corner on 10 minutes, while we tested Brighton keeper Christian Walton for the first time three minutes later, Jan Vertonghen’s 30-yard drive bringing a low save out of the stopper.
While Albion were concentrating on defensive duties early on, they threatened for the first time on 16 minutes when Rohan Ince lofted a pass over the top of the defence for Kazenga LuaLua. He was pushed wide as the ball skipped off the wet surface and lashed his shot well wide.
The visitors thought they should have had a penalty on 28 minutes when a deep cross was flicked away by Federico Fazio but appeared to strike Naughton on the hand. Referee Mark Clattenburg waved away their appeals as the loose ball fell to Lewis Dunk, who blasted over.
Shortly after, Lennon poked out a boot to Roberto Soldado’s cross but sent the ball over the bar, while Danny Holla’s 30-yard free-kick at the other end never troubled Michel Vorm.
We went close on 36 minutes, Naughton threading a delightful pass inside the defender for Soldado whose cutback fell to Kane. His shot was blocked, the rebound landed nicely for Soldado only for Walton to keep out his effort from a tight angle. Kane picked up the loose ball and delivered an inviting cross to the far post but Lennon fired his volley well wide.
Brighton had the last opportunity of the half, when Paddy McCourt made his way through the middle of the pitch and dinked a pass to Adrian Colunga in the area, but he was denied by a great challenge from Vertonghen as he shaped to shoot.
We made a change at the start of the second half with Lamela replacing Lennon and were on the front foot immediately, Kane’s glancing header just eluding Soldado’s dive in the six yard box and also the far corner of the net.
And the breakthrough finally came on 54 minutes with a delightfully worked goal. Davies drilled a ball into Soldado, he laid it off to Lamela, received the ball again and then slipped the Argentine in on goal, Lamela finishing neatly with his right foot past Walton.
It was so nearly 2-0 on the hour mark as Vertonghen led a breakaway and released Townsend, he advanced to the edge of the area and laid it off to Soldado on the right, but his effort crashed off the crossbar and went out for a throw in on the far side.
Then, in the 69th minute, the Spanish striker found himself in an almost identical position in the area, this time from Mousa Dembele’s pass, but Walton was equal to his effort.
Townsend so nearly found Kane inside the area after a mazy run through the Brighton midfield, his pass just a touch too heavy, but the pair were involved in the 74th minute as we doubled our advantage. Lamela led the charge forwards this time and released Kane down the left, he held up play before finding Townsend inside the area. The winger got to the byline, cut a ball back and Soldado tried a shot which hit defender Adam Chicksen and then keeper Walton, with Kane on hand to snaffle the rebound and find the back of the net.
The visitors rarely troubled our defence in the second period, but Vorm was alert enough in the closing stages to deal with a Colunga drive from the left channel inside the area as we saw out the game with the minimum of fuss.
Spurs (4-2-3-1): Vorm, Naughton, Vertonghen, Fazio, Davies, Stambouli, Dembele, Townsend (Chadli 81), Kane (Paulinho 84), Lennon (Lamela 46), Soldado. Substitutes (not used): Friedel, Dier, Chiriches, Mason.
Brighton (4-1-4-1): Walton, Calderon, Hughes, Dunk, Chicksen, Ince, Holla, Forster-Caskey (Gardner 65), McCourt (Teixeira 65), LuaLua (Mackail-Smith 88), Colanga. Substitutes (not used): Ankergren, Greer, O’Grady, Bennett.
Goals: Spurs – Lamela 54, Kane 74.
Yellows: Spurs - Fazio.
Referee: Mark Clattenburg.