Charlie Austin (left) celebrates after scoring Southampton's opening goal against Burnley at St Mary's Stadium in Southampton on October 16, 2016
It was a fine way for the Saints to start a run of seven fixtures in 22 days, with their next game a Europa League clash against Italian giants Inter Milan.
"I've got seven goals in the last six but we are seven unbeaten," Austin told Sky Sports. "That's the main thing. We knew it would be a frustrating afternoon if we didn't score early but once we scored one we kicked on.
"This team will always create chances, it's a dream for any striker. I am in the right place at the right time."
Claude Puel, Southampton's French manager, added: "I take pleasure this afternoon with a very good second half. Afterwards we came back with a good intention."
Burnley boss Sean Dyche felt his side should have had a penalty when the game was still goalless only for referee Mike Dean to take no action when Johann Berg Gudmundsson was halted by Virgil van Dijk.
"We knew they were a good side and in a good run of form, especially defensively," Dyche said.
"It's very difficult when you come to these places, particularly when we can't get a decision. I'm stunned by our penalty that wasn't given. I've seen it back and am absolutely amazed.
"They're a really good side and made it a really tough afternoon for us. But it's not helpful when it's 0-0 and you should have a penalty.
"Who knows after that? It changes the dynamic in the ground and the whole feel of the game."
Burnley kept the game goalless until the second half before the Saints scored three times in 14 minutes.
They might have been ahead earlier in the match but Austin's fifth-minute header was well-saved by Burnley goalkeeper Tom Heaton.
Southampton's Dusan Tadic was also denied by Heaton when one-on-one with the keeper in the 31st minute.
Redmond and Austin had chances in the first half but poor finishing kept the game goalless.
Southampton's pressure, however, told in the second period with a quickfire three-goal burst.
In the 52nd minute, Tadic's corner was headed across goal by Van Dijk and former Burnley striker Austin, at the second attempt and amid some poor defending by the Clarets, scored from in front of goal.
On the hour, Southampton doubled their lead courtesy of a fine goal from Redmond.
Following another corner, he waited for the ball to arrive before striking a low finish beyond the reach of Heaton and into the bottom right corner.
Minutes later, Austin put the result beyond doubt with a penalty, awarded after Gudmundsson brought down Sam McQueen.
Burnley managed a consolation goal 18 minutes from time when referee Mike Dean pointed to the penalty spot for a foul by Tadic on Ben Mee and Vokes scored as the Saints conceded for the first time in over 10 hours.
The Clarets' loss was their third in four games and left them just a point above the bottom three.