Lucas Moura's last-gasp hat-trick goal sealed a 3-3 aggregate draw, sending Spurs through on away goals to a clash with Liverpool in Madrid on June 1, leaving Pochettino in tears after the final whistle.
Pochettino acknowledged it may be difficult for his side to get back to reality against Everton but wants them to finish their league duties on a high, with a top-four finish not yet mathematically certain.
"We are in a moment, living a dream. It's difficult to be in the reality," said Pochettino, whose side will finish at least fourth unless they lose and Arsenal beat Burnley by a big enough margin to make up their eight-goal deficit to Spurs.
"It's our responsibility to create the atmosphere of competition because Sunday is an important day for us. To be ready is our first priority. To finish in the right way, in front of our fans, is the most important thing now.
"Of course, it was an amazing moment, one of the best we live all together after five years.
"I know very well it's difficult to stop to think after all we achieved on Wednesday, but our responsibility is to be ready.
"We're in a good position but it's not done. We need to take the game, be competitive with all energy and to finish in the right way will be a massive boost to prepare for the Champions League final against Liverpool in Madrid."
Pochettino will be without Son Heung-min and Juan Foyth on Sunday following their red cards in the defeat at Bournemouth, with Harry Kane, Harry Winks and Davinson Sanchez all still recovering from injuries.
Jan Vertonghen, Danny Rose and Victor Wanyama are also doubtful, and Pochettino is wary that Everton – who have won five of their past seven league games – will present a real threat to his depleted team.
"Everton have great momentum, very competitive and they finish the season very well," Pochettino said.
"They have very good players and it's normal that they're going to come here to compete. They had a clear week to prepare. We cannot be distracted, we need to be focused on our game."
Marco Silva, meanwhile, is not expecting Tottenham to be suffering a hangover from their European success.
"They did well, you can say they are living their dream. But they are professional, the Premier League is not done yet," Silva told reporters.
"They have one game to play, like us. Even if they enjoyed the last few days. It is possible not to be fully focused, but they will be ready. They will be full of confidence."