Son profited from some excellent hold-up play from striker Vincent Janssen to stroke Spurs into an early lead before cutting in from the left and firing a great effort across the goalkeeper and into the top corner a quarter of an hour later.
Middlesbrough were buoyed by some second-half changes and pulled themselves back into the game through Ben Gibson's header, but their late efforts came to nought as Spurs hung on.
Son slides Spurs into early lead
After an exciting start to the match which saw both sides with early opportunities, Spurs struck first through their South Korean star.
Son, who had started the game on the left, switched flanks to the right and cut inside where Janssen was holding the ball up inside the area before teeing the wide man up for a neat finish across Victor Valdes and into the bottom-left corner.
Spurs were bouyed by the goal and Dele Alli stung the palms of Valdes with a long-range drive from a central position with the hosts failing to make any real impact on the game.
Screamer doubles the advantage
A quarter of an hour after the opener, Son exemplified his quality with a stunning goal, this time from his normal left-flank position.
He cut in from the wing onto his right foot and let fly with a scorching shot which soared into the far top corner to send the visiting fans, on their second-longest league journey of the season, into raptures.
Later in the first half, Spurs should have had a third as the ball fell to Janssen inside the area but the Boro defence got back to cover.
The ball then ricocheted out wide to Moussa Sissoko, but Valdes smothered his effort at the near post to keep the home side with a fingertip on the game.
Visitors dominate against sluggish Boro
The game should have been put to bed shortly before half time. Arsenal loanee Calum Chambers, making his first appearance for Middlesbrough, dallied on the ball at the edge of his own area and was tackled by Christian Eriksen but Alli failed to capitalise on the chance, blazing over the bar as Valdes rushed out.
The visitors showed no sign of slowing down at the beginning of the second period. Jan Vertonghen stepped out of defence and combined with Alli to feed Janssen, but the Dutchman was stretching by the time he pulled the trigger and his first-time shot swerved high and wide.
Double substitution adds home impetus
With half an hour to play, Aitor Karanka tried to freshen things up by introducing Adama Traore and Jordan Rhodes from the bench, the latter making his first appearance of the season.
Traore, a product of Barcelona's La Masia youth academy, showed his intent immediately with a frighteningly quick surge through the centre which caught Spurs off guard, but his through ball for Rhodes was slightly over-hit.
Instantly, Boro's threat was transformed and they pulled a goal back within five minutes of the change.
Moments after delivering a shocking free-kick straight out of play, Stewart Downing put in a superb set-piece from the left wing onto the line of the six-yard box and Gibson rose above Victor Wanyama to direct a header back across goal and inside the far post.
Late pressure comes to nothing
With a quarter of an hour to play, Downing delivered another excellent ball from the left which Rhodes met powerfully wide a header from the edge of the area, but the forward couldn't direct it out of the reach of Hugo Lloris.
Spurs continued to pose a threat and Valdes had to get down well to hold onto a Janssen shot after neat interplay and some last-ditch defending inside the Boro penalty area.
The hosts had another chance to get back into the match in the last minute of normal time as a cross into the area was cleared up but not out, but substitute Adam Forshaw couldn't control his close-range volley.
Spurs saw out five minutes of stoppage time with little incident to claim another three points and leave Middlesbrough struggling in the bottom half after a third consecutive defeat.