Southampton had dropped to 18th position for the first time this season when Stoke City moved above them on Saturday.
Although one point from Sunday's game was not enough to move them any higher, Pellegrino believes it was another improved performance of the type they have shown since losing 5-2 to Spurs a month ago.
"I think we are on the right path," he said. "In the last four games the response has been good. They (Tottenham) are an amazing team with top players.
"We cannot transform the performance into goals and more points but we need to show again in the next game.
"Every single game for us now is massive.
"Confidence is sometimes not the best when you are in a bad moment and don't want to take risks. But little by little we improve."
Pellegrino is well aware that clubs like Crystal Palace, Everton, Leicester City and West Ham United have all improved by changing managers after falling into the bottom three this season.
A fighting draw gave him cause for optimism, however, despite coming under pressure a few months into the job with only one league win since October.
It is his first season at the club where Pochettino made his name and earned a move to Tottenham four years ago.
Since then Spurs have finished fifth, third and second, but they will be five points behind the four Champions League places if fourth-placed Liverpool win at bottom club Swansea City on Monday.
"To win the game we should do better and play better," Pochettino said of Sunday's performance. "It wasn't great.
"The game was even, we had more possession but we needed to be more aggressive and create more chances.
"You must give credit to the opponent, but I am not happy with our performance.
"We needed to move the ball quicker and create more, we made a lot of mistakes. It is impossible to win the game when you make these mistakes.
"It is two points we are going to miss at the end of the season because it is so tough between six or seven teams.
He did have encouraging words for his old club.
"It is temporary that Southampton are in the relegation zone, for sure they will push up the table," Pochettino said.
(Reporting by Steve Tongue, editing by Ed Osmond)
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