James Maddison’s moment of magic early in the second period earned a point at St Mary’s but the Foxes were unable to fully overcome a sloppy opening period.
Defender Jonny Evans initially fired the visitors level, in between first-half goals from Saints duo Jan Bednarek and Che Adams, while Harvey Barnes and Jamie Vardy went close to snatching an away win.
City boss Rodgers hailed his side’s resilience but was left frustrated by the slow start to the Premier League encounter on the south coast.
He said: “By the end, it was two points dropped. At half-time you might have taken the point.
“But second half, the reaction was much better.
“Like I’ve said before, there’s a great resilience in our team, a great fightback second half.
“First half, we didn’t taken on the burden of the game, we didn’t start well at all and that was the disappointment.
“It’s a lack of concentration, it’s a lack of aggression in key moments of the game and that’s what hurts us at times.
“The two goals were really poor goals to concede and it just comes from not being aggressive enough and a lack of concentration.”
Leicester have fallen agonisingly short of Champions League qualification in the past two seasons and cracking the top four this term already looks a tall order.
A point in Hampshire was sufficient to move them to eighth but left them with just one win from five as their inconsistent form continued.
Attacking midfielder Maddison, who was on target for the third successive game, collected the ball inside a crowded 18-yard box four minutes into second period and expertly cut inside the sliding Tino Livramento before lashing past Saints goalkeeper Alex McCarthy.
“Whenever we got him on the ball, he looked a real threat,” Rodgers said of Maddison.
“His goal was outstanding. He worked very, very hard, so very pleased for him.”
Southampton hold a five-point buffer on the relegation zone after scoring more than once in a top-flight fixture for only the third time this campaign.
Saints began brightly as they looked to respond to Saturday’s 4-0 thrashing at Liverpool and fully deserved their half-time advantage.
Yet they would have slipped to a third successive loss had McCarthy not expertly denied Barnes, before Vardy uncharacteristically blazed over after going through on goal 15 minutes from time.
Southampton boss Ralph Hasenhuttl said: “When you see the chances they had in the second half, they were the big ones.
“We deserved the lead in the first half so a point was fair.”
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in