Knowing a win would put them level with second-place Manchester United, Tottenham started nervously, but Lennon cut in from the right after getting on the end of Assou-Ekotto's brilliant crossfield pass and slotted past Tim Howard to settle the home crowd's nerves.
The full-back then added a second just after the hour when he blasted a 30-yard shot which flew past Howard off Tim Cahill's backside to seal the success.
The win marks another chapter in a remarkable resurgence for Spurs, who began the season with two hidings but have recovered to go on a run of one defeat in 18 games. They are now level with United and within three points of leaders Manchester City.
Indeed, Spurs have a real chance of cracking the Manchester clubs' grip on the league this weekend when they will move joint top if they beat Wolves, with City not playing until Monday.
Scott Parker's was again missing for Spurs due to a knee injury, and Everton - boosted by Cahill's inclusion - started brightly.
The Toffees could have gone ahead in the first minute had Louis Saha not shot wide.
It took 10 minutes before Emmanuel Adebayor had Tottenham's first chance, which he blazed over.
Luka Modric found Adebayor in the box in another Spurs attack, but his first touch let him down and Howard gathered.
Adebayor's touch let him down again in the 25th minute as he wasted a golden opportunity.
Gareth Bale galloped through after a brilliant ball from Rafael van der Vaart and squared to the unmarked Adebayor - but he miscontrolled.
Keen to make amends, Adebayor sprinted in to the box to collect Modric's flick-on and he went down claiming he had been pushed by Sylvain Distin. Martin Atkinson waved play on and replays showed he was right to do so.
Cahill's clumsy challenge on Modric had Spurs appealing for another spot-kick, but Atkinson gave nothing.
Van der Vaart curled a wicked shot on to the roof of the net in the 34th minute as Spurs upped the pressure and that pressure paid off a minute later when Lennon scored his fourth goal of the season.
The winger picked up Assou-Ekotto's cross-field pass, drifted past two Everton players and squirted a shot that beat the wrong-footed Howard.
Michael Dawson and Younes Kaboul both put in telling interceptions as Everton pushed for an equaliser before the break and Landon Donovan earned a yellow card in injury time for a late tackle on Assou-Ekotto.
Modric was cut down by Phil Neville just after the restart but Spurs played on.
With the Croatian lying on the floor, Bale picked the ball up five yards inside his half and charged at the Everton defence. They backed off but the Welshman's deflected shot bounced kindly in to Howard's arms.
With Modric back on his feet, Assou-Ekotto found Lennon at the back post with a clever pass but the winger headed wide.
Marouane Fellaini headed over and moments later was then lucky to escape a booking for using knocking Van der Vaart in the head with his forearm.
David Moyes' injury concerns grew soon after when Distin was replaced by Shane Duffy.
Spurs doubled their lead in the 63rd minute through Assou-Ekotto's first goal since the opening day of the 2009-10 season.
The full-back cut inside from the wing and after being given time by the Everton defence, cracked a 30-yard shot which deflected off Cahill and flew in.
Leighton Baines flung himself in front of Adebayor to deny the striker as Spurs kicked in to top gear.
A silly tug on Royston Drenthe's shirt saw Lennon enter the book before he was substituted for former Everton player Steven Pienaar.
Drenthe thought he should have had a penalty after he went down under pressure from Pienaar but Atkinson disagreed.
A clumsy tackle from Kaboul on Drenthe led to another penalty shout but again the referee waved play on, summing up Everton's night.