The Clarets, who sacked long-serving manager Sean Dyche last week, are in 18th place in the standings on 28 points from 32 games with Everton on 29 points with a game in hand.
Southampton should have taken an early lead when Oriol Romeu was left unmarked at a corner but his header was off target and Burnley were able to build themselves into control.
Burnley, with their Under-23 coach Mike Jackson and injured club captain Ben Mee in temporary charge of the team, were playing with a level of creativity and verve that they had struggled to find under Dyche this season.
There were more chances at both ends, with Romeu missing an almost identical chance for Southampton, before Burnley doubled their lead a minute before the break with Nathan Collins heading in a Josh Brownhill corner.
The Lancashire club host Wolverhampton Wanderers on Sunday and a win -- or even a draw -- would move them above Everton and out of the bottom three before Lampard's side face Liverpool in the Merseryside derby at Anfield later that day.
Goalscorer Roberts said there was no lack of confidence in the Burnley camp.
"There will always be belief. We know we have quality," Roberts said. "We will give everything to try and keep this club up. We have five games to give everything. Mike hasn't changed everything, but has given us a bit more freedom to do what we want."
Jackson finds himself uncertain as to whether he will be in charge for the visit of Wolves, but praised the way his players dealt with the unexpected changes at the club.
"The players remembered what they were. We haven't done loads different but they have come together as a group. Everyone has mucked in and it is a great result," he said.
"I will probably speak to the chairman and he will let me know if I am needed for next game. I am here to prepare the group, I don't want to think or talk about all that."
Southampton manager Ralph Hasenhuettl admitted his side had not done enough to win the game.
"We had the first big chance but couldn't take it. We were lacking everything after the first 15 minutes and it was not good enough," Hasenhuettl said. "In the end it was too easy for them to defend us, we were not dangerous enough.
"I gave them the chance to change things at the start of second half, when you start like this at Burnley it's tough to come back. We were far away from deserving something."