Spurs had failed to win their three previous league matches at their temporary home and were frustrated by Bournemouth's defensive setup in the first half. Eddie Howe switched to a three-man defence and packed his midfield to make his side as difficult to break down as possible.
Even so, the Cherries had the two best chances of the first half, as firstly Junior Stanislas had a shot blocked by Davinson Sanchez after Josh King's cutback. Hugo Lloris was then called into action when a Stanislas corner deflected off several bodies and towards the Spurs net, before the Frenchman's sprawling save.
Kane's only sight of goal in the first half was when he beat the offside trap to latch onto Eriksen's pass, but his tame shot was saved comfortably by Begovic.
The half-time nerves around Wembley were settled when Eriksen gave Tottenham the lead shortly after the break, exchanging passes with Dele Alli on the edge of the box and riding the weak challenge of Simon Francis before finishing left footed into the bottom corner.
Kane, who had scored 15 goals in his last 10 games for club and country, then wasted a couple of chances to give Spurs a cushion.
After heading home Kieran Tripper's cross from an offside position, the prolific striker went through one-on-one, only to be thwarted by Begovic. When Alli recycled the ball and squared it to Kane eight yards out, Nathan Ake made a crucial block.
Former Spurs hero Jermain Defoe nearly grabbed Bournemouth an equaliser off the bench, but his low shot on 82 minutes was saved by the feet of Lloris.
That was as close as Bournemouth came to snatching a point as they fell to a fourth successive away defeat which leaves them 19th in the table.
One of Tottenham's own substitutes, Georges-Kevin Nkoudou, suffered what appeared to be a nasty injury 40 seconds after his stoppage-time introduction as he clashed with Begovic while trying to score Spurs a second on the counter.
A single goal proved to be enough for Tottenham to open up a gap to fourth-place Chelsea, who lost to rock-bottom Crystal Palace today.