Mauricio Pochettino's side earned a slender advantage thanks to Son's clinical strike late in the second half at the Tottenham Hotspur Stadium.
Son scored the first goal in Tottenham's posh new abode against Crystal Palace last week and now the South Korea forward has bagged their maiden European strike in the arena.
While Son soaked up the plaudits, much-maligned keeper Lloris also enjoyed a sweet moment of redemption in a troubled campaign when he kept out Aguero's first-half spot-kick.
Tottenham's victory was an unexpected twist to an enthralling encounter, especially after Kane injured his left ankle in a tackle on Fabian Delph early in the second half.
But City had only themselves to blame for failing to earn an advantage for the second leg in Manchester on April 17.
Having won the League Cup, reached the FA Cup final and kept the destiny of the Premier League title in their own hands, this was City's next hurdle in the race for an unprecedented quadruple.
But City boss Pep Guardiola is adamant it is "almost impossible" to achieve a clean-sweep of all four major trophies.
He might be right on the evidence of a City display that lacked the cutting edge and intensity required to keep Tottenham at bay.
City's first defeat since a January 29 setback at Newcastle is not fatal to their European ambitions, but they will have to improve on this tired effort to stop Tottenham reaching their first European semi-final since 1984.
This was City's 52nd competitive match of a draining season and Guardiola made four changes from the team that laboured to a 1-0 win over Brighton in the FA Cup semi-final on Saturday.
Fernandinho was selected instead of Kevin De Bruyne, the defensive-minded Brazilian lining up in midfield in a move that showed Guardiola was willing to sacrifice a little of his attacking philosophy in deference to Tottenham's quality.
Tottenham have been riding a wave of excitement in their new £1 billion, 62,062-capacity stadium, while City wilted in the red-hot atmosphere at Anfield in the Champions League quarter-finals against Liverpool last season.
If Guardiola was concerned about subduing Tottenham and their boisterous crowd, his side were handed a golden opportunity in the 12th minute.
Raheem Sterling cut in from the left and left three Tottenham defenders in his wake before unleashing a shot that struck Danny Rose's raised arm as the Tottenham left-back slid in to block.
There were no penalty appeals from City, but after consulting VAR, Dutch referee Bjorn Kuipers awarded the spot-kick.
However, Lloris plunged to his left to keep out Aguero's penalty with a superb save.
Kane was embroiled in a physical battle with City defence's but his attempt to fight fire with fire proved his downfall in the 55th minute.
Crunching into a late challenge on Delph on the touchline, Kane appeared to twist his ankle before immediately signalling he would have to come off.
The sight of the England captain limping badly down the tunnel could have been fatal for Tottenham.
Kane, who has scored 24 times for Tottenham this season, is no stranger to ankle problems.
Yet Tottenham thrived in his absence for seven matches with left ankle ligament damage earlier this season.
And once again they rose to the challenge to snatch the lead in the 78th minute.
When Son beat the City offside trap, it appeared his touch was too heavy to take advantage.
But he alertly kept the ball in play, taking advantage of Delph's pause to protest the decision by slamming a low drive under Ederson from close range.
Guardiola once infuriated Pochettino by calling Tottenham 'the Harry Kane team' -- now Son has provided the perfect riposte to that jibe.