Stekelenburg endeared himself to the Goodison fans with his late double save and Koeman will have been happy with large parts of what he saw from his new troops.
The ex-Southampton boss, appointed in June as successor to Roberto Martinez, had claimed ahead of the game that his players were only at 70 per cent of the fitness he wanted.
They did appear to tire as time went on but there was certainly plenty to enthuse the Dutchman here as he looks to revive the Merseysiders after the disappointments of the Martinez era.
Even without John Stones, now a Manchester City player, they appeared full of energy as the contest got under way and swiftly took the lead when Tottenham allowed Barkley’s delivery to bounce across the box and into the net.
It was Spurs who were looking sluggish, with Harry Kane and Dele Alli – two of five players in the team who, unlike Barkley, had played in England’s humiliating Euro 2016 defeat to Iceland – struggling to make much of an impact.
A few days on from selling Stones, Everton showed the kind of defensive solidity that was rare under Martinez as skipper Phil Jagielka and new signing from Aston Villa Idrissa Gueye, among others, excelled in blunting Spurs’ efforts in attack.