After coming from behind to beat Watford in midweek, Spurs dominated their visitors for lengthy spells at Wembley but lacked a killer instinct in the absence of injured duo Harry Kane and Dele Alli.
Lucas Moura started in Kane's normal position but posed scant threat to a packed Newcastle defence, while at the other end Salomon Rondon and Ayoze Perez were a nuisance throughout.
Just as Newcastle looked set to extend their unbeaten Premier League run to three matches, Son's shot from the edge of the box squirmed under Dubravka and secured a win that keeps Spurs' faint title hopes alive.
Lucas should have given the hosts an early lead when he powered a free header harmlessly wide before Erik Lamela crashed a better effort onto the crossbar as Spurs began to turn the screw.
Newcastle's first chances came after the half-hour mark when Perez's shot rebounded to Rondon just outside the six-yard box, but the Venezuelan hesitated and his shot was crowded out.
Moussa Sissoko deftly met a high, looping cross with a side-footed volley that deserved a goal only for Dubravka to get down well to parry it away seven minutes before half-time.
Davinson Sanchez sent a header narrowly wide of Dubravka's net early in the second half and moments later Rondon went even closer at the other end, beating Toby Alderweireld in the air and hitting the post with his try.
Fabian Schar hooked off the line when Christian Eriksen thought he had flicked the ball into the net from close range, but Newcastle were eventually breached.
Son's low drive looked to be heading straight into Dubravka's arms yet it wriggled underneath his dive and found the net.
What does it mean? Newcastle's revival falters
Rafael Benitez might have sensed the chance to upset a another team with Premier League title aspirations after his side beat Manchester City in midweek, but Dubravka's error cost the Magpies a precious point and kept Spurs in the hunt for silverware.
Rondon leads the line
While Tottenham missed the presence of Kane in their attack, Newcastle benefited from having a very clear target man to aim at in the form of Rondon, who caused problems all afternoon.
Eriksen fails to inspire Spurs
Eriksen's creativity is all the more important to Spurs in the absence of the injured Alli but he was a shadow of his industrious best, giving Lucas and substitute Fernando Llorente little to work with.
Key Opta Facts
- Tottenham have lost just one of their last 27 home Premier League games in February (W19 D7).
- Newcastle have lost 13 of their last 14 Premier League games away to ‘big-six’ teams (D1), since beating Tottenham 2-1 in December 2015.
- Son has been directly involved in 14 goals in his last 10 appearances in all competitions for Spurs (nine goals, five assists).
- Llorente has been involved in seven goals in his last eight appearances for Tottenham in all competitions (five goals, two assists).
- Llorente provided assists in back-to-back league games for the first time since March 2008 for Athletic Bilbao.