The visitors fell behind to an early Ross Barkley free-kick at Goodison Park, in what was Ronald Koeman's first competitive game in charge of Everton, but Pochettino's side came on strong after the interval, with Erik Lamela's header restoring parity.
Tottenham would have secured all three points if not for two marvellous saves from Maarten Stekelenburg late on, but overall Pochettino was pleased with his side's performance and the result, especially considering they were second-best in the first half, with only Everton's wastefulness in front of goal stopping them falling further behind.
"I am very happy with how we handled the game and how we finished it, but I am very disappointed with how we started," Pochettino said. "We were slow and we conceded the early goal, and I was very unhappy. In the second half we looked like the team we were last season, so we need to analyse that, but it is only the first game of the season.
"At half-time I told them that we had made a lot of mistakes, and if you cannot play in possession and create chances you will not get anywhere. But in the second half there was a lot more energy and we always had different options and ways to play.
"We had a lot of players involved in the Euros so it is difficult to be fit and ready, but we can take a lot of positives from today.
"I was tough with them but all the managers do that when you are not happy. It is normal. We were very close to winning the game."
One of the aforementioned saves from Stekelenburg came when the Dutch goalkeeper reacted instinctively to parry away Vincent Janssen's close-range effort, and Pochettino, who also confirmed that Hugo Lloris would undergo scans on a hamstring injury sustained in the first half, was quick to praise his close-season recruit, who sparked life into Tottenham's lacklustre attack when he was introduced from the bench.
"I thought Vincent was excellent," Pochettino added. "He came on and brought good energy, and looked to get in behind Everton, it helped us."