Granit Xhaka's brilliant finish sent Mikel Arteta's side ahead in the 11th minute, but Stuart Armstrong levelled for Southampton in the second half, and the points were ultimately shared.
Arsenal remain top of the Premier League table, two points clear of second-placed Manchester City, while Southampton sit 15th, boasting 12 points from their opening 12 games of the season.
Arsenal took control of the contest in the early stages, with Southampton goalkeeper Gavin Bazunu called into action to deny Gabriel Jesus and Xhaka, before Martin Odegaard fired wide of the home side's post, with all three chances coming inside the first nine minutes.
As a result, it was not a surprise to see the Gunners make the breakthrough, with Xhaka finding the back of the net with an emphatic finish after meeting a low cross from Ben White.
Jesus came close to doubling Arsenal's advantage in the 25th minute, but he fired wide from a tight angle, before the Brazilian also forced Bazunu into a smart save late in the half.
The visitors were comfortable at the start of the second period and had a half-chance on the hour through Jesus, but Mohamed Elyounoussi did well to make the block.
Southampton were level in the 65th minute, though, with a smart team move ending with Elyounoussi finding Armstrong, who brilliantly placed the ball past Aaron Ramsdale.
The Saints then came close to a second shortly after through Romain Perraud, but his effort was just too high, as Arsenal came under serious pressure for the first time in the contest.
The visitors thought that they had regained the lead in the 79th minute through Odegaard, who found the back of the net with a fine finish, but the ball had just crossed the line for a Southampton goal kick before Kieran Tierney could deliver.
Both teams had opportunities to carve out openings late on, but neither could find the back of the net, with the points ultimately being shared at St Mary's.
Next up for Arsenal is a trip to PSV Eindhoven in the Europa League on Thursday before welcoming Nottingham Forest to North London in the Premier League next weekend.
Meanwhile, Southampton will be bidding to build on the positive result against the division leaders when they travel to Crystal Palace in their next match.