Ross Barkley's free-kick caught everyone out and gave Everton the lead, only to be pegged back by Erik Lamela's powerful header in the second half.
It was lively from start to finish, but Everton have Maarten Stekelenburg to thank for several point-saving stops.
Barkley gives Everton an early lead
As Koeman made his official bow as Everton boss, he couldn't have wished for a better start. Spurs were going in hard on the Toffees and that presented plenty of free-kick opportunities. It may not have been conventional, but a Barkley delivery managed to evade everyone and slide past Hugo Lloris at the far post to open the scoring.
Everton were electric in the opening stages and could have added to their lead, had it not been for Lloris keeping out Gerard Deulofeu's strike. It required an even better save to keep out Phil Jagielka's towering header, as the French stopper scuttled backwards to get finger-tips to the ball and prevented it from creeping under the bar.
Dull end to the half
Both Kevin Mirallas and Christian Eriksen then exchanged chances, but neither truley tested Lloris and Maarten Stekelenburg, respectively. Everton were doing a superb job of keeping Harry Kane under wraps, which seemed to continually frustrate the North London side.
Towards the end of the half, Mauricio Pochettino made the decision to call Michel Vorm from the bench to replace Lloris. Whether it was injury or not wasn't clear - perhaps Euro fatigue - but the stand-in stopper did superbly well to stay big and deny Deulofeu from grabbing a second before the break.
Lamela heads Spurs level
The second half began rather quietly, though Tottenham were much improved after the half-time break. Both sides had chances to add to the tally, but Delle Alli skied from the edge of the area and both Mirallas and Deulofeu - who were still very dangerous - had long-range efforts held by Vorm. Vincent Janssen was also introduced early on.
Spurs' pressure told on the hour mark as a stylish move down the right paid dividends. After good work down the wing the ball was set for Kyle Walker to cross, and his wicked delivery was met by the on-the-move Lamela to bullet the ball past a helpless Stekelenburg. Suddenly, the mood had changed at Goodison.
Aside from a Ramiro Funes Mori pot shot in the latter stages, the final half hour was totally dominated by Pochettino's side. Stekelenburg had been solid throughout the game but made a stunning point-blank save from Janssen's poke at goal, before tipping Lamela's deflected shot onto the crossbar.
It was a case of what could have been for both sides, as contrasting showings in either half showed their weaknesses and strengths in equal measure.