Stoke goalkeeper Asmir Begovic notched the opener in just 13 seconds at the Britannia Stadium, with a long kick forward taking one bounce and flying in over Artur Boruc's head.
But Saints equalised when Jay Rodriguez headed home three minutes before the break before Pochettino's side had two penalty claims refused in the second period as the game ended all square.
He told Sky Sports News: "We were very unlucky, we were 1-0 down in the first 15 seconds so it was hard to take that.
"But after that we were prepared, we showed that we had good character, good personality and we did well after that."
Regarding the goal, he added: "I just think it was an unlucky play, it was a very complicated match overall.
"I think the pitch was not up to par today - the wind was very strong but I'm not here to make any excuses.
"It was just a very tough match, a very tough day at the office and we did our best today, we showed very good character."
Pochettino believed his side created enough chances to win the game, adding: "I think we controlled the match, especially in the second half.
"We created a lot of chances in the second half. There were two possible penalties - I'm pretty sure they were penalties - and we created the last chance of the match as well.
"So overall, I'm really pleased with the resilience and the character and the identity that my team showed."