Sam Byrne (2) and Conor McAleny did the damage with Byrne's deflected strike to double Everton's lead in first half injury time a particular blow after the teams were evenly-matched in the first half.
But the game was essentially over five minutes into the second half as Byrne and McAleny put the Toffees 4-0 up.
To their credit, the lads found a response in the Merseyside rain but it was a long journey back to Hotspur Way on a Bank Holiday Monday.
First-team squad regulars Harry Winks and Josh Onomah provided the experience as we lined-up for our first match in the competition.
Both teams created chances in an entertaining first half. Luke Amos had our first sight at goal on four minutes, his shot deflected wide before Tom Glover did well to block from Byrne.
Everton were handed the chance to take the lead on 14 minutes when a somewhat innocuous challenge from Amos nevertheless tripped Callum Connolly right on the corner of the penalty area. Glover got a firm hand on David Henen's spot-kick but couldn't keep it out.
McAleny and Joseph Williams forced Glover into routine saves before Shayon Harrison's persistence opened it up for Emmanuel Sonupe to test Joel Robles for the first time on the half-hour.
Our best chance of the first period came via a wonderful cross from Anthony Georgiou from the left on 37 minutes. Sonupe did well to get across his man but saw his header flash just wide with Robles motionless on his line.
Everton made it 2-0 in the second minute of added time as Byrne was found at the far post from a corner, his first shot was parried straight back to him by Glover and his second attempt brushed off Christian Maghoma and nestled in the far corner.
If that was a blow just before the break, then we were left reeling by a quick one-two minutes into the second half.
Henen got to the right byline before finding McAleny with time and space to fire home for 3-0 on 47 minutes and just two minutes later, Ryan Ledson played in Byrne and he fired past Glover from the left angle for 4-0.
We responded quickly though as Maghoma powered home a header from Winks' corner on 52 minutes and you felt there was still belief in the players that they could drag themselves back into the game.
Our best spell followed and Robles had to be at his best to touch away Kyle Walker-Peters' effort from the right angle on 56 minutes before beating away Sonupe's fierce drive from a similar position on the hour.
That was the closest either team came to adding to the scoreline until the final seconds when substitute Charlie Owens fired across Robles, only to see his effort cannon back off the post.
We next take on FC Schalke at Stevenage on Tuesday, November 3 (7pm) before concluding our group against FC Porto, also at the Lamex Stadium, on Wednesday, December 23 (also 7pm).
We play each team in the group once and, upon completion of the group stage, the winners and runners-up from each pool will progress to the quarter-finals, semi-finals and final. These are all to be played as single-leg ties.
Everton U21s: Robles, Pennington, Connolly, Ledson (Walsh 76), Browning, Holgate, Williams (C), Grant (Yates 86), Byrne, McAleny, Henen (Evans 67). Substitutes (not used): Hewelt (GK), Charsley, Graham, Donohue.
Spurs U21s: Glover, Walker-Peters, Lesniak (C), Maghoma, Georgiou, Amos (Stylianides 55), Goddard (Owens 67), Winks (Walkes 55), Sonupe, Onomah, Harrison. Substitute (not used): Voss (GK).