Chelsea responded last weekend after Spurs had trimmed their lead, beating Bournemouth 3-1, but Antonio Conte's men face a less than straightforward trip to Manchester United on Sunday.
"Our job was to get three points and we got three points," said Spurs manager Mauricio Pochettino.
"Now if they (Chelsea) fail, we're there. Now the pressure is on Chelsea to try to win at Old Trafford."
Spurs appear the only team capable of denying Chelsea the title and will get a chance to measure themselves against the leaders in next weekend's FA Cup semi-final at Wembley.
Kane, 23, is the first Spurs player to have scored at least 20 league goals in three successive seasons since the great Jimmy Greaves in the late 1960s.
In another milestone, Spurs have now won 12 consecutive home games in the top flight within the same season for the first time in their history.
While Bournemouth remain seven points above the relegation zone, they will watch this weekend's remaining games involving the teams below them with a degree of anxiety.
"We know we need more. We've known all along," said beaten manager Eddie Howe.
"The danger is if everyone says, 'You're safe.' We've got to focus the players' minds that we're not."
Making his first start in five weeks after an ankle problem, Kane was quickly in the thick of the action with a volley that was blocked by Steve Cook.
Dembele also looked in the mood, unleashing a fierce effort from Kyle Walker's lay-off that was parried by Artur Boruc, and in the 16th minute he put his side in front.
- Wilshere injury -
The goal stemmed from an error by visiting captain Simon Francis, who allowed the ball to roll behind in the mistaken belief a goal kick would be awarded.
Instead a corner was indicated and from Christian Eriksen's in-swinging delivery, Dembele took a touch before ramming home his first goal since January 2016.
Three minutes later it was 2-0.
Arsenal loanee Jack Wilshere, lustily booed by the Spurs faithful, saw a pass blocked by Eric Dier and Kane's clever flick freed Son to scamper in and drill a shot between Boruc's legs.
Bournemouth flickered, Cook heading over at a corner and Benik Afobe threatening to apply the finishing touch to a Junior Stanislas cross, but Spurs were soon back on top.
Boruc was extended by an effort from Eriksen and had to field a shot from Son, while Eriksen also thrashed over with his left foot.
The half-time interval brought only temporary respite for Bournemouth, who found themselves a further goal behind within three minutes of the restart as Kane scored his 13th goal in 12 league games.
Walker clipped a pass into Kane's feet and he took Francis out of the game with a Cruyff turn before spearing a left-foot shot past Boruc.
Boruc saved from Kane three times in nine minutes as the England man continued to torment the Bournemouth rearguard.
Wilshere appeared to injure his ankle in a challenge on Kane and had to make way for Lewis Cook, the home fans' gleeful catcalls accompanying him as he limped down the tunnel.
Boruc was also tested by Eriksen, Son and Ben Davies as Pochettino urged his men to keep attacking from the touchline.
Bournemouth belatedly registered a shot on target in the 74th minute through Charlie Daniels, but Spurs had the last word as goal-shy substitute Janssen bundled in from a stoppage-time corner.