Spurs had survived a major scare in the previous qualifying stage last week, coming from behind to beat nine-man Lokomotiv Plovdiv with two late goals in Bulgaria.
And the trip to North Macedonia threatened to prove similarly troublesome when Valmir Nafiu cancelled out Erik Lamela's fifth-minute strike.
A visiting team including debutant Joe Hart and the returning Dele Alli toiled for a time until Son Heung-min - fresh from a four-goal salvo at Southampton - restored the lead and then turned provider for Harry Kane, whose goal ensured a play-off with Maccabi Haifa.
Lamela's opener looked to have teed up a straightforward evening for Tottenham, found completely free by Son on the right side of the area to slide a calm finish inside the bottom-left corner.
Spurs dominated but did not threaten again until Toby Alderweireld headed narrowly wide in the 26th minute, with Kostadin Zahov then saving Alli's scruffy eight-yard effort.
Further Tottenham openings followed in the second half without causing Zahov undue concern and Shkendija were level against the run of play after 55 minutes.
Ljupco Doriev fizzed a pass right to Nafiu, who arrowed a sensational effort high past Hart into the top-right corner.
Kane, Giovani Lo Celso and then Lucas Moura were called from the Spurs bench, yet the England captain followed Lamela in squandering a big headed chance with the goal gaping.
Son was clinical when his chance came, though, controlling as Zahov parried Lucas' shot and firing into the roof of the net.
Kane created each of Son's four goals in their previous Premier League game, but the roles were reversed as Tottenham made sure of their place in the next round, a cross from the left met with a firm downward header.
What does it mean? Spurs forced to turn to stars
Having repeatedly bemoaned Tottenham's tough schedule, Mourinho would have been frustrated to see a team missing Kane, Lo Celso and Lucas fail to get the job done.
He was therefore forced to call on the rested trio just three days before the visit of Newcastle United. It paid dividends, but the starting XI - Son apart - did themselves and their coach few favours.
Son the Spurs star
Given the focus Mourinho is placing on the Europa League, Son's display here might even be considered more important than his stunning performance against Southampton.
His poise in the area had already prompted the breakthrough goal for Lamela, but two more telling contributions were required in the final 20 minutes to advance. The winger created four chances in all.
Mourinho said this week Alli - back in action for the first time since his half-time substitution in the defeat to Everton - held "99 per cent" of the responsibility for the improvement he required.
The coach did not appear overly enthused by this display, expressing his frustration with Alli on a couple of occasions, as the midfielder missed a big first-half chance and won just one of his six duels while failing to register a tackle or interception before he was replaced by Kane.
Spurs must now travel back to London ahead of hosting Newcastle at Tottenham Hotspur Stadium on Sunday.