The hosts were the better team for long periods and took a deserved lead through Gaston Ramirez's well-taken 59th-minute goal.
However, they were pegged back by Andy Carroll's deflected free-kick seven minutes later and must now wait before being able to confirm their Premier League status for next season.
"We are a bit disappointed," Pochettino said. "We had the feeling we could win the game, but we weren't able to.
"We put in a big effort to get the three points, but we conceded very quickly after we scored. We were unlucky because the ball [Carroll's free-kick] went through [Steven] Davis's legs.
"I do think we deserved the victory over the 90 minutes, but you cannot deserve victories, you have to fight for them."
Pochettino claimed the foul that led to Carroll's goal was wrongly given against defender Jos Hooiveld and also lamented the wet and windy conditions that he felt stifled his side.
However, he praised his players for nullifying West Ham's strengths and overcoming a virus that swept through the camp in the build-up to the game.
He added: "Jos said that it was not a foul, but these things happen in football. It is hard to control that.
"It was hard to play on a pitch that was so wet and against a rival that was so physically strong. Overall, during the 90 minutes we were able to nullify the way they played the long ball. We defended really well against them.
"We were patient in the game, but it is a shame that we conceded a goal really quickly after we scored. It was unfortunate, but that is football.
"I am very happy, especially because during the week we had problems with a virus that meant the players were not able to train properly."