The hosts were hampered by a supporter boycott, which contributed to their lowest crowd of the season, and went behind to Nacer Chadli's 25-yard shot.
Jack Colback scored after the break, but Spurs replied through Christian Eriksen's free-kick before Kane struck.
Spurs moved into sixth, while Newcastle suffered a sixth straight defeat.
That worrying sequence equals Newcastle's worst run of results in the Premier League as they remain seven points above the relegation zone.
Only second-from-bottom QPR have a worse points record in 2015, and with a trip to Loftus Road and third-bottom Leicester still to come, John Carver's side could yet be caught.
They rarely troubled Spurs, who have hardly been in top form themselves and are seven points adrift of fourth-placed Manchester City.
But a fourth win in 12 games lifted Mauricio Pochettino's side above his former club Southampton, who they visit next Saturday - the Argentine's first return to St Mary's since leaving last summer.
A return to winning ways for Tottenham and a late boost for English striker Kane contrasted with the mood engulfing St James' Park.
Reports suggested that up to 16,000 fans would boycott this match in protest at the way the club is run by owner Mike Ashley.
And although those numbers failed to materialise, there were patches of empty seats as the attendance reached 47,427 - almost 5,000 down on the capacity.
Those who stayed away did not miss much from a Newcastle team whose most lively outlet was Ayoze Perez, still searching for a goal in 2015.
The visitors did not impress either early on, and there looked no immediate threat before they took the lead on the half-hour. Chadli picked the ball up in a central area and as the Newcastle defence retreated, the Belgian drilled low for his 11th of the season.
The hosts were unable to muster a response until shortly after the interval when Perez broke down the right, cut the ball back and after Daryl Janmaat's shot was blocked, Colback tucked in the rebound.
Finally the home supporters found their voice but their optimism lasted seven minutes. Colback fouled Eriksen after the Dane broke down the left, and the Spurs midfielder curled in the resulting free-kick which found its way into the net without a touch.
Nabil Bentaleb headed over soon after, and a relatively quiet Kane was denied by a brilliant save from Tim Krul.
But the 21-year-old was not to be denied, racing through during a late counter-attack to become the first Spurs striker to reach 30 goals since Gary Lineker in 1991-92.
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