A dreadful performance from Liverpool's back four and goalkeeper Simon Mignolet helped Kane make it eight goals in his last six Premier League games as Spurs moved level on points with second-placed Manchester United.
Kane's first league goal at Wembley and a Son Heung-min strike ensured Tottenham raced into a two-goal lead inside 12 minutes.
Mohamed Salah briefly got Liverpool back into the game, but Dele Alli netted for first time in six league matches before half-time to keep Spurs in control.
And Kane struck again to seal a statement victory early in the second half, Spurs' first win in 10 Premier League meetings against the Reds.
This damaging defeat in front of a record Premier League crowd leaves Liverpool ninth in the table, 12 points behind leaders Manchester City after a troubling start to the season for Klopp.
Liverpool have leaked 16 goals in nine games – their worst top-flight defensive record at this stage of a season since 1964-65.
The first shot of the match produced the opener inside four minutes, in-form Kane striking for the hosts with a goal for which Liverpool could only blame themselves.
Dejan Lovren was flat-footed in dealing with Kieran Tripper's throughball, allowing Kane to get in behind and take advantage of a poor decision by Mignolet - who raced off his line - by taking it past the goalkeeper and applying a left-footed finish from 12 yards.
It was soon two, with Spurs keeper Hugo Lloris starting a quick counter-attack with a long throw that Lovren completely missed as he attempted to head clear on the halfway line, giving Kane the chance to cross for Son to guide a first-time, left-footed finish past Mignolet – the South Korea international's first top-flight goal since May.
It could have been worse for Liverpool, as Son fired a powerful effort off the crossbar after a Christian Eriksen pass opened up the struggling away defence once again.
Liverpool threatened a comeback in the 24th minute when Kane lost possession and an excellent pass from Jordan Henderson released Salah, who outpaced Serge Aurier and Jan Vertonghen down the right channel before scuffing a right-footed finish in off the far post.
At the other end, Mignolet raced out to deny Son with his legs after a fine Alli pass and Klopp had seen enough of his ineffective back-line, taking off Lovren to introduce Alex Oxlade-Chamberlain in a tactical re-shuffle.
But Spurs struck again in the third minute of first-half stoppage time, Joel Matip only heading a deep Eriksen free-kick as far as Alli, who volleyed in from just inside penalty area to secure a deserved two-goal lead at the interval.
After 56 minutes, the rampant Kane inflicted further damage, smashing in a rebound from six yards after Mignolet had flapped at Tripper's free-kick prior to an effort from Vertonghen being cleared off the line by Roberto Firmino.
Liverpool almost responded when Philippe Coutinho's left-footed curler from 20 yards was clawed out of the top corner and onto the bar by Lloris.
Lloris then had to keep out decent efforts from James Milner and Salah to ensure there were no late nerves in a fourth consecutive league win for Spurs, who will hope there are no ill-effects for Kane after he came off in the final minutes holding the back of his leg.
Key Opta Facts:
- Mauricio Pochettino’s side scored more goals with just 14 shots in this game (4), than they had in their four previous Premier League games at Wembley (three goals from 90 shots).
- Tottenham’s second goal was the 1000th that Liverpool have conceded in the Premier League (now 1002), making them the sixth team to have reached this milestone (Tottenham 1237, Everton 1225, Aston Villa 1186, West Ham 1163 and Newcastle 1148).
- Harry Kane’s opener was his first Premier League goal at Wembley Stadium, with his 29th shot at the ground. In the last eight games in which he’s scored for Tottenham in all competitions, Harry Kane has scored at least twice.
- Only Robbie Fowler (6) scored more goals than Mohamed Salah (5) in their first nine Premier League appearances for Liverpool.
- This was only the third time Liverpool had conceded four goals in the opening hour of a Premier League game – Stoke City in May 2015 and Southampton in February 1994.
- Since his debut for Liverpool in August 2013, Simon Mignolet has made more errors leading to a goal than any other player in the Premier League (13).