COVENTRY City manager Mark Robins insisted their poor first half performance is what cost them against Saints on Tuesday evening.
Kyle Walker-Peters and Che Adams combined to give Saints the lead before Adams doubled his tally just before the break.
Coventry's Haji Wright had missed from the spot at 0-0, the US international slipping before sending his effort onto the crossbar.
Jake Bidwell, who played at Swansea under Russell Martin, cut Southampton's lead but the hosts held firm to secure an important win.
“It would have been a different story if Haji didn’t slip, but we still didn’t compete," Robins admitted post-match. “We didn’t arrive until halftime.
"We have to decide if we are going to be challengers or not. In the first half, we decided that we didn't want to be challengers.
"We let them score two easy goals. I’m told the first one is handball, that's unfortunate if it is. I will have to take a look but, ultimately, there is nothing I can do about it.
“The penalty miss has an impact. If we score that the crowd is edgy. That would have given us a foothold, but we didn’t lay a glove on them for 45 minutes. That is what has cost us the game.”
Southampton looked more threatening again after a poor showing at Blackburn Rovers on Saturday, where they registered just two shots on target.
Martin elected to deploy a more traditional back four, likely as a result of Coventry's lack of press, something which frustrated Robins.
“They are a good team. They have really good players and they move the ball quickly. Their rotations are really good.
"Unless you get pressure on them and limit those rotations, they will cause you problems. We were much better at negating that in the second half. The threat was still there.
“But the switch was on too often. We were not getting pressure on them enough.”
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