The result gives Jose Mourinho's men a disappointing start to the new season, while Carlo Ancelotti's side will be buoyed by the impressive result after a number of new signings in the offseason.
The match was goalless at half-time, with Everton, spurred on by new star James Rodriguez, especially unlucky to find a breakthrough in the first 45 minutes.
But it didn't take long for the visitors to grab a deserved lead after the break. From a set piece near the left touchline, forward Dominic Calvert-Lewin rose to meet Lucas Digne's cross and fired home a thumping header from the middle of the box to put the Toffees up 1-0 in the 55th minute.
The goal was his 20th league strike since the start of 2018-19, which is second most among English players 23 years or younger, trailing only Marcus Rashford's 27.
It was the first time Mourinho had lost the opening match of a Premier League season.
"You look at the quality that's come in, it's an instant uplift," Calvert-Lewin said. "As a centre forward you look at the creativity -- they've played at the top level. There should be chances this season and I need to be there."
Tottenham got an early scare when Ben Davies played a horribly stray pass across field and Richarlison gathered it and rounded Lloris, only to stumble and fire over an empty goal.
It at least provided a wake-up alarm for Tottenham who suddenly burst into life with Harry Kane just failing to get on the end of Son Heung-min's cross, Dele Alli having a shot tipped over by Pickford who then blocked Doherty's close-range effort after a brilliant lobbed pass by Kane.
Rodriguez, signed from Real Madrid this week and looking to rediscover the heights that made him one of the world's most sought-after players after the 2014 World Cup, offered a performance that augurs well for the season.
He was at the heart of much of Everton's best moves and almost bagged a goal when curling just wide before halftime.
He also set up Richarlison after the break but again the Brazilian fluffed his finish.
Tottenham will play a Europa League qualifying match midweek before visiting Southampton in Premier League play on Sunday. And Everton will host West Ham on Sunday, three days after an EFL Cup second-round match at Salford City.
Information from Reuters was used in this report.