Gaston Ramirez fired them ahead in the second half.
That lead lasted just seven minutes, though, as Andy Carroll struck home via a slight deflection off Steven Davis to earn a share of the spoils.
"We had a feeling we could win the game, but were not able to," Pochettino said. "We put in a big effort to get the three points and conceded very quickly after we scored.
"It was hard to play on a pitch that was so wet against an opponent that are physically very strong, so overall I think we did a very good job against them.
"I congratulate my players because they did quite well in conditions that actually favoured the rivals' style of play.
"Even though the conditions favoured them, we actually played well. I think we played the game as we wanted and should have. We imposed our brand of football.
"I think we defended well, nullified their style of play and cancelled them out.
"It is just a shame about the result because our performance was really good."