The big news in the buildup to the clash was the absence of club captain Pierre-Emerick Aubameyang once again due to 'personal issues'.
Head coach Mikel Arteta made two changes from the side that comfortably dispatched Newcastle United last time out, with Nicolas Pepe and Hector Bellerin coming in for Aubameyang and Kieran Tierney.
Arsenal fashioned the first chance of the game when Lacazette was played though by Granit Xhaka in the first minute, but Alex MacCarthy was quick off his line to save a certain goal. However, the Frenchman should have done better.
Two minutes later, the hosts went ahead through Stuart Armstong, with the Southampton midfielder scoring on the volley from a James Ward-Prowse corner kick.
However, Arsenal were not behind for long as Pepe showed great technique to level the scores after he had been played in by Xhaka.
The rest of the first half was played at an upbeat tempo, with both sides fashioning good chances, but the two goalkeepers proved equal to the task.
With the game looking likely to be level at the break, a brilliant sequence of play helped Arsenal take the lead in the 38th minute.
A killer through pass by Lacazette split the Southampton defence, allowing Saka to run straight at MacCarthy.
The Arsenal forward showed great composure to round the goalkeeper, before slotting a right-footed shot into an empty net to send the visitors into a half-time lead.
The second half saw the hosts up the tempo, with good chances falling to Danny Ings and Theo Walcott. However, Bernd Leno proved up to the task for everything thrown at him.
Arsenal made sure of the result when Lacazette bundled the ball over the line after an excellent centre by Saka. The goal came at the expense of him smashing his leg against the post but he would not have cared much.
The victory moves the Gunners up to eighth on the Premier League table and here, we shall be having a rundown of five talking points from the fixture.
#5 Arsenal enact revenge for FA Cup ouster
Arsenal saw their FA Cup defence end at the 4th round, when a first half own-goal by Gabriel Magalhaes condemned them to a defeat at St Mary's.
That loss came just four days ago, and the Gunners would have had revenge on their mind heading into the weekend's Premier League fixture.
When Armstrong put the hosts ahead inside three minutes, it might have seemed like a case of lightning striking twice in the space of a few days.
However, a resilient fightback form the Emirates outfit ensured that they took all three points back to north London.
It also saw the Gunners avenge their FA Cup defeat on Saturday, and fans of the club will be hoping that things get progressively better from here.
#4 Southampton's recent poor run in the league continue
A good start to the season saw Southampton soar up the table, with the highpoint coming when the Saints topped the standings for the first time in 32 years after a 2-0 victory over Newcastle United in November.
However, Ralph Hasenhuttl's side have failed to match their early-season form and came into the clash with Arsenal on a run of just one win from their last six Premier League games.
Credit to the Hampshire side for going toe-to-toe with the Gunners, but despite giving their all, they ended up on the losing side.
The defeat means that they failed to usurp their nearest rivals on the table and remain in 11th spot. They could lose further ground on the table at the conclusion of the gameweek, which is a far cry from the early season highs.