Lamela, who scored twice in the Europa League last week, came off the bench at half-time and burst on to Roberto Soldado’s pass before firing past keeper Christian Walton on 54 minutes.
Kane, who went one better than Lamela against Asteras Tripolis in Europe, grabbing a hat-trick, tapped home a second from Andros Townsend’s deflected cross after Soldado had smashed a shot against the crossbar.
A much-changed Spurs side bossed the game, with Brighton’s best chance falling to Kazenga LuaLua in the first half, but his shot sailed high over the bar from a tight angle.
Only defender Jan Vertonghen survived from the 2-1 home defeat at home to Newcastle on Sunday, which has seen Tottenham widely criticised for a lack of passion and mental strength.
But Maurico Pochettino’s team, with 10 changes, seemed determined to prove their doubters wrong and Kane, Aaron Lennon and Vertonghen all had shots on target blocked in the opening period.
LuaLua may have done better when he latched onto a long diagonal pass on 16 minutes and the same could be said for Danny Holla, who failed to hit the target with a well-placed free-kick from distance.
Lamela replaced Lennon, who appeared to have a knock, at the break and he immediately looked threatening before finding only his fourth goal for the club since a £28million move from Roma in 2013.
Brighton had a decent shout for a penalty when Federico Fazio looked to have fouled LuaLua inside the box, but it was waved away by referee Mark Clattenberg, who gave the same reaction to Soldado’s protests shortly after.
But Kane put the game beyond the visitors and extended his tally as Spurs’ top-scorer this season – tapping home his seventh goal of the campaign and fourth in three games.