The pair combined for a tenth goal this season, as Spurs snatched a scarcely deserved win through Son's 76th-minute header. It was the South Korean's eighth league goal of the season, with all but one assisted by Kane, speaking to their remarkable understanding.
Though the combination was familiar, the manner of the goal was not, with Kane arching to head Erik Lamela's corner towards the back post where Son was the most alert to pounce and beat Nick Pope with just Tottenham's second shot on target.
The Portuguese, however, had played down comparisons between Tottenham and his record-breaking Real Madrid side, who scored 121 goals in La Liga in 2011-12, and said Spurs are still searching for the right "balance" between attack and defence.
Sure enough, the visitors were both more compact than in the final eight minutes against West Ham but far less exhilarating than in the opening 16 minutes of last weekend's wild 3-3 draw.
Mourinho's line-up looked designed to contain, with the workmanlike Lucas Moura starting ahead of Steven Bergwijn or Gareth Bale, Pierre-Emile Hojbjerg and Moussa Sissoko protecting the back four and Ben Davies and Matt Doherty preferred to Serge Aurier and Sergio Reguilon at full-back.
Spurs did contain Burnely...but did little else either side of Son's instinctive headed finish. Until his assist, Kane's biggest contribution was clearing James Tarkowski's header off the line.
As Mourinho expected, the Clarets were physical, compact and committed defensively but the visitors' approach suggested their manager was more scarred by the end against West Ham than inspired by the beginning, when Spurs played on the front foot and tore through the Hammers at will.
Ultimately, Mourinho's XI and his side showed the Clarets too much respect, particularly given Spurs' historic struggles at breaking down sides who defend deep and in numbers.
Even Mourinho's substitutions appeared frustratingly conservative, with Lamela replacing Lucas on the hour, while Bale and Carlos Vinicius watched on.
The Argentine, though, did make a difference, adding more impetus and precision in the final third and playing his part in the goal.
Given Spurs' improvement this season, particularly in the final third, the display felt in danger of being a backwards step and an almost unnecessary reminder that Tottenham remain a work in progress. As it was, it felt like a massive three points, lifting Spurs into fifth place with a third consecutive away win.
Spurs full-backs disappoint as Mourinho searches for balance
A significant factor in the Tottenham's pedestrian approach was the lack of penetration from full-backs Ben Davies and Matt Doherty.
Mourinho recalled the pair after starting Reguilon and Aurier against the Hammers last weekend, presumably to add more defensive nous to his side.
With Hojbjerg and Sissoko offering protection in the middle of the park, Spurs needed their full-backs to get forward and attack Burnley, but neither Davies nor Doherty offered anything in the final third.
Indeed, Spurs sorely missed the pace of Reguilon and Aurier, who are arguably more of a liability defensively but love to run into space in the opposition's half.
More than any other position, Mourinho's full-backs seem to reflect the 'balance' he is seeking. Go too attacking with Aurier and Reguilon and his side are in danger of squandering cheap free-kicks and being hit on the break. Go too conservative and they suffer in the final third.
Mourinho shows ruthless side
Mourinho could barely cobbled together a side on Tottenham's last visit to Burnley back in March but now the manager's selection dilemmas extend way beyond the first XI.
Absent from the match-day squad at Turf Moor were Paulo Gazzaniga, Serge Aurier, Japhet Tanganga, Davinson Sanchez, Danny Rose, Gedson Fernandes, Harry Winks, Steven Bergwijn and Dele Alli, who started 146 Premier League matches between them last season, speaking to the new depth of Mourinho's squad.
Aurier, Sanchez and Winks were the three players arguably most to blame for West Ham's comeback last weekend and it felt like no coincidence that they were missing from the 18 altogether. With so much quality at his disposal, Mourinho can now afford to be ruthless.
Lloris 6, Doherty 5, Alderweireld 7, Dier 6, Davies 4; Sissoko 5, Hojbjerg 5, Ndombele 6 (Lo Celso 79' 6); Lucas 5 (Lamela 57' 7), Kane 7, Son 7 (Rodon 94' 6).
Star man: Harry Kane, 7. Decisive at both ends, clearing James Tarkowski's effort of the line and assisting Son again.