Despite starting with Kane, Son Heung-min and Gareth Bale in attack, Jose Mourinho's side delivered a largely flat attacking performance against the Baggies.
Indeed, they needed a good Hugo Lloris save and the intervention of the left-hand post in the second half to prevent Slaven Bilic's men taking the lead.
However, Kane secured a fourth win in a row away from home in the top flight when he headed Matt Doherty's delivery beyond Sam Johnstone.
150 – Only Alan Shearer (212) and Sergio Agüero (217) scored their 150th Premier League goal in fewer appearances than @HKane, who has done so this afternoon in his 218th match in the competition. Cometh. #WBATOT pic.twitter.com/5Rjb8iLRgu— OptaJoe (@OptaJoe) November 8, 2020
The best of the early chances fell to Son, who had lots of time and space in the penalty area but eventually saw a shot blocked over by Semi Ajayi.
West Brom twice came close to a breakthrough, Karlan Grant nodding a good Darnell Furlong cross wide and Eric Dier heading well off the line with Grant lurking.
It took until the hour mark for Spurs to register a shot on target, Sergio Reguilon stinging the palms of goalkeeper Sam Johnstone with a speculative strike from outside the box.
Giovani Lo Celso, on for the injured Tanguy Ndombele, had a great chance to make a swift impact but hooked his shot wide from barely 10 yards out.
Conor Gallagher clipped the post from the edge of the box and Lloris reacted superbly to tip away a Grant header following a corner, as West Brom's best chances went begging.
Johnstone thought he had secured a share of the spoils, saving well low to his right to keep out a strike from Carlos Vinicius, but his indecision as Doherty swung in a high ball allowed Kane to head in the winner.
What does it mean? Kane milestone goal sends Spurs top
Mourinho's side jumped to the top of the table and knew they would stay there for the international break if Leicester City and Liverpool failed to win later on Sunday. Regardless of that outcome, their unbeaten run in the league now stands at seven, while West Brom remain winless and on just three points from eight matches.
Kane, who reached 200 goals in all competitions for Spurs this week, is on 150 in the top flight in only 218 appearances. Only Alan Shearer (212 games) and Sergio Aguero (217) reached that tally quicker.
Ajayi a giant for Baggies
Bilic was deeply unhappy when West Brom offloaded Ahmed Hegazi, but Ajayi has since found some fine form. The Nigeria centre-back was outstanding against Kane, Son and Bale, with a team-high five tackles, four interceptions, three blocks and 11 possessional gains.
Bale scored the winner against Brighton and Hove Albion last week, but he offered little in his first league start since rejoining Spurs, mustering just two shots and attempting only 21 passes, the fewest of any of Spurs' starting XI.
Key Opta stats
- Tottenham have won each of their opening four away games of a league season for just the fourth time in their history (also 1949-50, 1960-61 and 2017-18).
- West Brom have failed to win any of their first eight games of a league season for only the second time, having last done so in 1985-86, when they would go on to be relegated from the top-flight.
- Tottenham have kept a clean sheet in back-to-back away league games for the first time since October 2018.
- West Brom have failed to score in back-to-back home Premier League matches for the first time since a run of four games in a row from April to May 2017.
- Kane has been directly involved in 27 goals in his last 14 starts for Tottenham in all competitions, scoring 17 and assisting 10.