Having suffered a poor start to the season, Arsenal have been firing on all cylinders lately and Nicolas Pepe, Bukayo Saka and Alexandre Lacazette all got on the scoresheet after the visitors fell behind to an early goal.
"I think we were good from the start. We conceded a goal from a corner but responded straight away. We were good on the ball, we got out of trouble and made some big chances," Arteta told BT Sport.
"Our pressing was brilliant. We were good on the break. It was a strong performance."
Bukayo Saka celebrates scoring for Arsenal
(POOL/AFP via Getty Images)
"It is tough to win here, they are strong and have been in form. It showed a lot of intelligence and resilience from the team in the way they managed the game," Arteta added, before sounding a note of caution.
"We have still a lot to do. We have to improve, be humble and train good for the next game," he said ahead of their home fixture with Manchester United on Saturday.
Southampton coach Ralph Hasenhuettl was disappointed with how his side gave up their early lead and gave a blunt assessment of their shortcomings.
"It was easy to score against us today, we didn't close the gaps between centre-back and full back. Such an opponent can kill you, that is why we didn't deserve anything today," he told BBC Sport.
"We had moments to score. We tried the best we could today but it was not enough," he added.