Palace had their best chance at the end, with Kagisho Dikgacoi firing straight at Hugo Lloris.
Soldado, 28, scored 24 goals in 35 appearances for Valencia last season and had the perfect start to his Premier League career.
Tottenham duo Kyle Walker and Lennon consistently linked well down the right - and their running eventually proved decisive just after half-time.
Lennon found space in the area but his cross clattered into Moxey's outstretched hand. Tottenham did not win a penalty in the whole of the 2012-13 season but referee Mark Clattenburg pointed to the spot and Soldado kept his cool, slotting home as Speroni dived the wrong way.
Since 1 January 2013, this is only the second time Spurs have won when Gareth Bale has not scored. The previous game was against Stoke on 12 May.
Palace frustrated Tottenham in the first half, with Andre Villas-Boas's team reduced to shooting from distance.
Mousa Dembele fired over the bar from 25 yards and Gylfi Sigurdsson's clipped effort from a similar distance was tipped over by Speroni.
Palace's passing was particularly wayward and they needed a set-piece to create their first chance on target, as Aaron Wilbraham headed straight at Lloris from Owen Garvan's free-kick.
With half-time approaching, Walker played through a scuttling Lennon, who evaded the defenders, only for Speroni to thwart Soldado's attempt.
Crystal Palace 0-1: Tottenham: Villas-Boas on deserved victory
The best chance of the first half came from the final piece of action. Lennon chipped a cross to the back post, where new signing Nacer Chadli duly headed the ball into the crowd.
In the second half, Palace boss Ian Holloway made a treble substitution after Tottenham's goal, bringing on Jonathan Williams, Marouane Chamakh and veteran Kevin Phillips.
Sigurdsson should have sealed the match but side-footed wide after Walker had found Soldado in the penalty area.
Moments later, 40-year-old Phillips sprung the offside trap and almost found himself in on goal. Lloris raced out to intercept Dikgacoi's ball over the top, sliding outside of his box and only just managing to hack the ball away.
Tottenham did not win a penalty in the whole of the 2012-13 season
Palace looked a different side after the changes, with Phillips a constant menace and Dwight Gayle finally free to show some pace up front.
Phillips had another good chance but his shot dribbled wide from 25 yards.
Substitute Jermain Defoe had an opportunity to seal the match, racing past four Palace players, only to shoot wide.
Palace then had their best chance as the ball dropped to Dikgacoi in the area, but the South African could only shoot directly at Lloris.
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