The Spurs duo struck in the last 20 minutes as Mauricio Pochettino's side put one foot in the group stages of Europe's second-tier competition.
It was far from straightforward for Tottenham in Larnaca, though, and the Premier League outfit were perhaps lucky to be only trailing by one goal at half-time.
Adrian Sardinero gave the hosts a 14th-minute lead, but they could not capitalise on further chances to extend their advantage.
The influential Kane then hit the woodwork with 20 minutes to go and it was a warning sign that AEL did not heed.
Soldado first levelled proceedings in the 74th minute before Kane also converted an Erik Lamela assist to give Tottenham victory.
Pochettino previously claimed he had "no interest" in the Europa League while in charge of Southampton, but with a UEFA Champions League spot now up for grabs for the winners, he is aiming to go all the way in the tournament.
They did not make a bright start, though, as Soldado missed the target from six yards before AEL took the lead.
Paulinho was caught on the ball inside his own half, and as a result, the ball broke for Sardinero, who quickly sped through Tottenham's defence before producing a clever finish into the bottom-right corner.
Pochettino's men failed to produce an immediate response and were nearly two goals down 11 minutes later, when Lukasz Gikiewicz nodded Sardinero's cross just over the bar.
Kane, one of few positives for Spurs in the first half, tested AEL goalkeeper Karim Fegrouche twice in quick succession after the half-hour mark, but the Cypriot side continued to look more threatening.
And Sardinero went close again eight minutes before the break, darting in from the right before he forced Hugo Lloris into a fine save as Spurs ended the half on the back foot.
Both sides struggled to get going in the second half and, despite the introduction of Nacer Chadli and Mousa Dembele, Spurs looked bereft of creativity.
But Lamela – signed for a fee in the region of £25million from Roma last year – changed the game after coming off the bench, as, shortly after Kane was denied by the bar, the Argentinian set up Tottenham's equaliser.
His clever run through the middle drew in defenders and he followed it with a sublime pass that played in Soldado, who coolly hammered a first-time effort in.
And the second swiftly followed – just six minutes later – with Lamela playing a pivotal role again.
This time Lamela came in off the left and found Kane, who beat an opponent before firing past the helpless Fegrouche.
Lamela almost got his name on the scoresheet after a solo run, but Fegrouche was on hand to deny him, as Spurs had to settle for a one-goal margin, a result that looked unlikely for the majority of the match.
The second leg will be played at White Hart Lane in seven days.
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