The lads forged into a 2-0 lead inside 30 minutes at the Lane, Everton hit back before the break but didn't look close to an equaliser until a couple of nail-biting moments in the final few minutes.
Indeed, such was the control central defenders Mills and the inspirational Charlie Lee had over the dangerous Everton front men James Vaughan and Victor Anichebe that it took the visitors until the 82nd minute to register a noteworthy effort at goal in the second half.
Everton then went close twice but there was little doubt the right team progressed to the last eight.
We took the lead after seven minutes as Mills timed his run to perfection to meet O'Hara's inswinging corner.
As the tackles flew in, Everton claimed a penalty when Anichebe tumbled under Lee's challenge but appeals were waved away and the visitors were cursing their luck further after 20 minutes when Vidarsson struck the woodwork from 20 yards.
Play switched end to end as Vaughan volleyed just wide and Gallagher darted out to save at Barcham's feet.
The second goal arrived on the half-hour and was a classic. Dawkins was flattened on the edge of the box and O'Hara curled a delightful free-kick into the corner.
Everton still looked dangerous though with Anichebe and Vaughan a handful and they got back into the game five minutes later. There was a touch of luck about it as Lee's crunching 50-50 squirted wide to Anichebe and he had time to cross for Hopkins to beat Forecast and head home.
We enjoyed our best spell at the start of the second half.
Craig Gallagher did well to keep out Andy Barcham's close-range effort three minutes into the half, Charlie Daniels whistled a shot inches wide from Barcham's clever lay-off and then Gallagher stood tall to deny Barcham one-on-one.
The intensity of the tie was maintained throughout but there was then a quiet period in both penalty boxes as the visitors looked for inspiration to break us down.
They finally created a chance eight minutes from time when lively substitute Kissock worked a yard in the box and drilled just wide and then, two minutes later, goalscorer Hopkins missed the chance of the night. With all Spurs' hearts in mouths, Anichebe nodded down to the midfielder but he somehow poked wide from six yards.
It needed just one more Tommy Forecast save from a deflected shot in injury time and that was that.
GUIDE TO THE ACTION
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4mins: First effort, Lewis blasted over after short corner routine between O'Hara and Dawkins.
7mins: GOAL. O'Hara's inswinging corner to near post flicked home by Leigh Mills.
Spurs 1, Everton 0
15mins: PENALTY? Anichebe onto bouncing ball, chased by Lee, went down under challenge but appeals waved away.
16mins: Vaughan volleyed wide from Hopkins corner.
18mins: Gallagher out quickly to save at feet as Barcham looked to latch onto Dawkins' flick.
20mins: WOODWORK. Vidarsson crashed ball onto crossbar from 20 yards.
24mins: Lovely cushioned pass from O'Hara to Barcham in left channel, low cross missed by defender, Dawkins taken by surprise and didn't react to chance.
28mins: Pacy Vaughan burst onto through ball, into box, overran it slightly, Forecast out bravely at his feet and clattered in 50-50 challenge.
28mins: YELLOW. Vaughan booked for challenge.
30mins: GOAL. O'Hara curled free-kick into top corner from 20 yards.
Spurs 2, Everton 0
35mins: GOAL. Forecast beaten to Anichebe's cross from right by Hopkins who headed home.
Spurs 2, Everton 1
43mins: Hopkins 25-yarder taken comfortably by Forecast.
Injury time: YELLOW. Wright for foul on Vidarsson.
Half-time: Spurs 2, Everton 1
48mins: SAVE. O'Hara's wicked inswinging free-kick missed by first defender and caused panic, Barcham onto loose ball, stabbed it towards goal from close range but Gallagher alive to make block.
50mins: CLOSE. Lovely move, Wright into Barcham who flicked it to supporting Daniels, took early shot from 25 yards that flew inches wide.
53mins: SAVE. Great control from Barcham, teased defenders before working a yard onto his right foot, attempted to dink home but Gallagher out quickly to block.
59mins: YELLOW: Phelan for foul on Barcham.
68mins: Ifil one-two with Maghoma, left foot shot from 16 yards grasped by Gallagher.
82mins: CLOSE. Kissock finally worked an opening for Everton, quick feet in the box and a low drive inches wide with Forecast at full stretch.
84mins: MISS. Anichebe knock down left Hopkins with easiest of tasks from six yards but he somehow prodded chance wide.
86mins: YELLOW. Wright booked for dragging back Hamed.
88mins: YELLOW. O'Hara for timewasting.
Injury time: SAVE. Forecast parried away Kissock's deflected shot.
Full-time: Spurs 2, Everton 1
Spurs: Forecast, Ifil, Wright, Lewis, Lee, Mills, Maghoma, Dawkins (Hamed, 77), Barcham, O'Hara, Daniels. Subs: Thyer, Button, Riley, Davis.
Everton: Gallagher, Wright, Wilson, Boyle, Hughes, Kearney, Hopkins, Vidarsson (Phelan, 46), Vaughan, Anichebe, Fowler (Kissock, 67). Subs: Jones, Holt, Molyneux.
Mr D Whitestone
Goals from Leigh Mills and Jamie O'Hara toppled Everton and booked our place in the quarter-finals of the FA Youth Cup on Tuesday night.