Pochettino's men had much the better of the first half at St Mary's and led at the break thanks to a 21st-minute header from Jose Fonte.
Goals from Olivier Giroud and Santi Cazorla in the opening seven minutes of the second half put Arsenal ahead, but Adam Lallana converted Jay Rodriguez's cutback just two minutes later to frustrate the London side.
Mathieu Flamini was sent off for the visitors for a two-footed lunge on Morgan Schneiderlin in the 80th minute, but although Southampton pressed hard in the final stages, they could not claim the winner that Pochettino felt they merited.
"I thought the result was good but we deserved to win the game," he said.
"We deserved a bit more. Overall, I want to congratulate the players because they played really well.
"The team was fantastic in the first half, but also in the second half when we had to get back into the game after conceding those two goals. The team was really brave and courageous and played really well to get back into the game.
"That’s the thing about football, you can make mistakes, and when you make mistakes against a team like Arsenal it's a heavy price to pay.
"We felt positive and optimistic about the future, I think we can continue to play well from this moment on."
‘Maybe I’ll be signed by Wrexham, who knows? English football would suit me – if an offer came, I wouldn’t think twice, I’d go immediately’ Euro 2024 cult hero reveals his UK dream
‘Managing Leeds? It was an option that appeared, but it wasn’t the right timing. I decided it wasn’t a good idea to leave the club I was at mid-season’: Premier League boss admits to turning down opportunity to replace Jesse Marsch in 2023
