Richarlison spurned a golden chance to put the Toffees ahead in the first half, but the visitors would open the scoring just after the break.
The only goal of the game came courtesy of Dominic Calvert-Lewin, who rose high to nod home a Lucas Digne free kick.
Check out 90min's player ratings from this opening day clash below.
1. Goalkeeper & Defenders
Hugo Lloris (GK) - 6/10 - Despite Everton's periods of dominance, the Frenchman had a quiet afternoon. Could do little about the goal.
Matt Doherty (RB) - 7/10 - His side's most potent attacking threat. Asked some questions of his opposite number and very nearly scored before the break.
Toby Alderweireld (CB) - 6/10 - A pretty clumsy performance. Should have been punished by Richarlison in the first half which set the tone for an error-strewn display.
Eric Dier (CB) - 6/10 - Out-jumped by Dominic Calvert-Lewin for the opener. Also conceded a few silly fouls.
Ben Davies (LB) - 6/10 - The Welshman was given little license to attack by Jose Mourinho. Also struggled defensively against the impressive James Rodriguez.
Pierre-Emile Hojbjerg (CM) - 5/10 - An pretty meh debut. Didn't do too much wrong but didn't do too much right either. Picked up a yellow card late on.
Harry Winks (CM) - 6/10 - Everton's good pressing gave him very little room to pull to pull the strings. Withdrawn after an hour having done very little.
Dele Alli (AM) - 5/10 - Did not get much joy during his 45 minutes stint. Finished terribly when he was finally prevented with a chance.
Lucas Moura (RW) - 5/10 - Lots of energy but very little execution. Became increasingly less clear where he was playing as the game progressed.
Harry Kane (ST) - 6/10 - Tepid by his own high standards but did gift Doherty a sight on goal with a delightful scoop pass. it was beautiful.
Son Heung-min (LW) - 6/10 - His poor decision making early cost Harry Kane an opening day goal.
Moussa Sissoko - 5/10
Steven Bergwijn - 5/10
Tanguy Ndombele - N/A
5. Goalkeeper & Defenders
Jordan Pickford (GK) - 7/10 - Saved brilliantly from Doherty in the first half and pulled off several other smart stops.
Seamus Coleman (RB) - 7/10 - Son kept him busy, but he dealt with most threats without breaking sweat.
Michael Keane (CB) - 8/10 - Composed performance. Targeted by Kane but stood up to the test well.
Yerry Mina (CB) - 7/10 - A welcome return for the former Barcelona man after he missed out on the final few games of last season. Imperious in the air and distributed nicely.
Lucas Digne (LB) - 7/10 - Sumptuous free kick led to the only goal of the game. One of many pinpoint deliveries provided by the left-back.
Allan (CM) - 7/10 - Looked slightly sluggish in transition but did not put a foot wrong in possession and never gave the Spurs midfield a moment's rest.
Abdoulaye Doucoure (CM) - 8/10 - An all-action performances at the heart of midfield. A couple of heroic tackles and drove the Toffees forward gallantly.
Andre Gomes (CM) - 6/10 - The least impressive of Everton's three midfielders. Kept the ball, back lacked incision in his passing.
James Rodriguez (RW) - 8/10 - Everton's creative fulcrum. More performances like these and he'll soon become a cult hero at Goodison Park.
Dominic Calvert-Lewin (ST) - 8/10 - Leapt like a freshwater salmon to open the scoring and looked very cool doing it - as he always does.
Richarlison (LW) - 6/10 - Provider an early contender for miss of the season in the first half. Spurned several more chances after that as well, but his quality was evident.
Gylfi Sigurdsson - 6/10
Moise Kean - N/A
Tom Davies - N/A