Victor Valdes then did well with a dive to his left to palm a Deli Alli effort wide of the goal, but he could do nothing to deny Son on seven minutes as he hit home from 10 yards out after a neat lay-off from Janssen with his back to goal.
But Spurs held firm and the visitors marched on with another three points.
Son Heung-Min was pleased with his brace, but feels Spurs should do better with converting their chances.
Kane will not be part of the visiting team as he is now nursing an ankle injury after damaging ligaments during a 1-0 win over Sunderland on September 18.
The England striker suffered an ankle ligament injury last week leaving a hole in the Tottenham attack and a player on the fringes last season is proving himself ready to reliev.
"They need to feel that we are in a project [where] we can win everything", he said. He picks up a pass from Janssen before slotting it past the helpless Valdes. He wins it on the left-hand side of the area before curling it into the top corner.
Edinburgh has turned Gillingham into one of League One's top sides
Karanka also addressed reports that Boro's 23-year-old centre-back Ben Gibson is line for a call up to the England senior team. Adama Traore and Jordan Rhodes, who will make his Premier League debut, will replace the frustrated Negredo and ineffective Ramirez.
Mauricio Pochettino's Tottenham Hotspur have confirmed their starting line-up and substitutes to take on Middlesbrough in an away Premier League fixture this afternoon.
A free kick from Stewart Downing was met by Gibson who headed across Hugo Lloris and in. He meets a deep cross from Downing but he can't get enough on the connect to glide the ball past Lloris.
Son made only 13 league starts last term, but now has four goals from his three top-flight appearances this season.
The South Korean worldwide has now scored six goals in his last six Premier League appearances, which is in stark contrast to his form when he first joined having gone 14 games without a goal prior to that.
Bournemouth have never beaten Everton in six previous attempts in all competitions (D1 L5), with their only draw coming in this exact Premier League fixture last season.