The victory continued Arsenal's recent resurgence. The Gunners failed to win in five Premier League games before Christmas but have now won five in a row to climb to eighth in the standings.
Pepe levelled proceedings with a close-range finish after being played through by Granit Xhaka's well-weighted through-ball.
Saka nudged the visitors in front on 39 minutes when Southampton goalkeeper Alex McCarthy rushed out of his goal but was rounded by the Arsenal man who shot into an empty net.
And Saka then set up Lacazette to give Arsenal a two-goal cushion in the 72nd minute with a close-range tap-in.
Southampton's high-pressing style paid dividends in terms of winning possession in dangerous areas, but their problems in front of goal continued and Bernd Leno made some fine saves to keep them out.
James Ward-Prowse had a late chance from a free kick but Leno made another excellent save.
Teenage winger Saka was delighted to get his fifth Premier League goal of the season as his side made amends for the defeat at the weekend.
"It was much better than Saturday, Saturday was disappointing. We are the (FA Cup) holders and we took it hard. We exploited their weaknesses and got the win we deserved," he told BT Sport.
Having suffered a poor start to the season, Arsenal have been firing on all cylinders lately, but manager Mikel Arteta cautioned there is plenty of work left to do.
"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."
It was a marked change from Arsenal's display at the same venue last Saturday, where Saints dumped them out of the FA Cup.
"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."