Gylfi Sigurdsson's corner was flicked on by debutant Cenk Tosun, finding the head of Wayne Rooney, and although his glancing effort beat Hugo Lloris, the linesman already had his flag raised.
The Tottenham striker scored eight goals in six matches, including back-to-back hat-tricks against Burnley and Southampton in a vintage month.
Kane worked hard, dropping deep to bring centre-backs Phil Jagielka and Mason Holgate out of position and that created space for Pochettino's players to run into.
However, it did not take long for Tottenham to change that statistic and, on 47 minutes, Kane tapped home after smart work by Son on the left.
The Baggies secured their first top-flight win in 21 games with a 2-0 result against Brighton and Hove Albion on Saturday. I feel it's easier for Turkish players to go play overseas now than it was in our time.
Spurs eased in to the fourth round of the FA Cup with a 3-0 victory over AFC Wimbledon last weekend but their focus will be firmly on Premier League matters tomorrow.
It hardly mattered, however, as Kane extended Spurs' lead moments later with a close-range finish over Pickford from Eric Dier's precise pass into the box.