Arsenal climbed to fifth in the Premier League table following this victory on the south coast, just three points behind Manchester City and four behind Liverpool — and the Reds have played a game more than the Gunners.
This, though, was not a performance to fill Gunners supporters with confidence as Arsenal stuttered and spluttered on a night when Alexis Sanchez bailed them out.
In a match devoid of any real quality it was the Chilean who provided two moments of magic to win the game. First he brilliantly found the net on 60 minutes after a sumptuous dummy, before he later turned provider for Olivier Giroud’s late header.
It means that with three games to go Wenger’s goal of keeping his perfect record of making the Champions League in every one of his full seasons in charge is still on the cards, if one of Arsenal’s rivals slip up.
“Every game we try to give ourselves a chance,” Arsenal goalkeeper Petr Cech said.
“We shot ourselves in the foot against Tottenham and we are aware we need to win all the remaining games to be in the race (for the Champions League).
“We had a big game against United and then here, where historically we haven’t done that well in the league.
“It was tough in the first half but then we found the opening and deserved to win.”
Arsenal’s away form coming into this trip to Southampton would have had them venturing to the south coast with a sense of dread. They had lost five of their last six away league games — as many as they had in their previous 27 trips.
They had to contend too without centre-back Laurent Koscielny, who was the only change from the side that defeated Manchester United 2-0 on Sunday as he missed out with a calf injury.
Southampton made just one change too from the team that held Liverpool on the same day, with Nathan Redmond coming in for Sofiane Boufal.
And it was the hosts who started brighter as James Ward-Prowse side-footed over the bar from outside the box after a matter of minutes.
Arsenal were trying to find their way into the match with their usual methodical approach but as they prodded and probed, mainly down the right through Alex Oxlade-Chamberlain, they found little joy as neither goalkeeper was really troubled in the opening 20 minutes.
Arsenal, though, were the side beginning to take control.
However, for all Arsenal’s possession, Saints were posing the real threat and it was they who should have taken the lead on the half-hour mark. Dusan Tadic found himself with space in the box, but his effort was blocked by Shkodran Mustafi before Cech saved Manolo Gabbiadini’s rebound with a smart save.
Arsenal were struggling to test Fraser Forster and their problems were made worse as their best attacking threat so far, Oxlade-Chamberlain, went off injured 10 minutes before the break.
Instead Cech was doing his best to keep the visitors on level terms and he was at full stretch to stop Nathan Redmond’s 25-yard drive arrowing into the top corner moments before the end of the first-half.
Arsenal came out for the second-half knowing their performance had to considerably improve if they wanted to leave Southampton with the three points that would keep their top four hopes alive, and it seemed as though those exact words from Wenger were ringing in their ears as they started urgently.
Aaron Ramsey, who had been one of the few bright sparks in the first-half, poked an effort just wide following the restart as the Gunners signalled their intent to attack.
Finally Arsenal were beginning to come to life and when the conjured some real quality on the hour-mark they duly took the lead. Mesut Ozil slid in Sanchez and the Chilean finished with aplomb, twisting and turning to leave Southampton defenders on the deck and the ball in the back of the net.
It was the moment of quality a drab and dour game had been crying out for and suddenly Arsenal’s confidence was flowing.
Arsenal, however, needed to be wary of being caught out and Ward-Prowse’s narrow header wide with 20 minutes to go was a remainder of how slender their lead was.
Bertrand, who had supplied the cross for Ward-Prowse’s chance, was the man causing all the problems and he delivered another pin-point ball to the head of Gabbiadini — but the Italian fluffed his lines.
However where Gabbiadini failed, Giroud succeeded to wrap up the three points for Arsenal with seven minutes remaining.
Again it was Sanchez providing the quality, delivering a perfect ball to the back-post, which Ramsey nodded into the path of a gleeful Giroud.
Forster 6; Cedric 7, Stephens 6, Yoshida 6, Bertrand 8; Romeu 7, Davis 6; Ward-Prowse 6 (70 Boufal 5), Tadic 6 (80 Rodriguez 5), Redmond 7 (70 Long 5); Gabbiadini 5.
Hassen, Caceres, Pied, Hojbjerg.
Cech 6; Holding 7, Mustafi 8, Monreal 6; Oxlade-Chamberlain 7 (36 Bellerin 6), Ramsey 7, Xhaka 7, Gibbs 6; Sanchez 8, Ozil 7 (88 Coquelin 5); Welbeck 5 (80 Giroud 7).
Ospina, Gabriel, Iwobi, Walcott.