Nemanja Matic pulled one back for Chelsea before the interval, but Naismith completed his treble in the closing stages to ensure the Blues, with three defeats and only four points from their five matches, made their worst start to a season since 1988.
Chelsea's second successive loss means they will trail Manchester City by 11 points by the end of the day if the unbeaten leaders win at Crystal Palace later on Saturday.
Mourinho's misery started in the 17th minute when Brendan Galloway whipped a cross towards Naismith, who advanced unchecked into the six-yard box to bury his header past Chelsea goalkeeper Asmir Begovic.
Naismith was only on the pitch due to an ninth minute injury suffered by Muhamed Besic, but he made the most of his opportunity by doubling Everton's lead with a fine low strike in the 22nd minute.
Matic reduced the deficit with a superb effort that flashed past Tim Howard from 30 yards in the 36th minute, but Naismith struck yet again in the 82nd minute to cap an embarrassing afternoon for the visitors.
Later, in the day's most eagerly anticipated match, Manchester United face bitter rivals Liverpool at Old Trafford.
United are likely to include goalkeeper David de Gea for the first time this season after the Spaniard signed a new contract this week following the collapse of his proposed transfer to Real Madrid.
The hosts could also give a debut to young French striker Anthony Martial, who arrived from Monaco for an initial £36 million on transfer deadline day.