However, instead of wallowing in mourning for their former leader, the home players put in one of their best performances of the season in front of a bustling Turf Moor atmosphere.
Story of the match
Caretaker boss Michael Jackson made two changes from the Burnley team that drew 1-1 draw with West Ham United. One of them was enforced, with the injured Ashley Westwood being replaced by Josh Brownhill. Roberts was also preferred to Matt Lowton.
By contrast, Ralph Hasenhuttl rewarded the starters who beat Arsenal at St Mary's by naming an unchanged eleven.
And it was the visitors who made the brighter start, penning Burnley into their own defensive-third while patiently probing.
The set-piece specialist James Ward-Prowse curled a delightful 7th-minute corner onto the head of an unmarked Oriel Romeu, who couldn't hit the target from ten-yards.
Burnley had another lucky escape less than 60 seconds later when Armando Broja miscued a shot in the centre of the box after the Clarets had sloppily rescinded possession.
It was as though the Clarets were a little bit tentative in coming into the post-Dyche era in front of the home supporters, however, they soon woke themselves up and brilliantly fired themselves ahead.
Brownhill did well to work the ball to Roberts on the edge of the box and the right-back shifted inside before curling a sublime left-footed effort into the far corner.
The goal rocked Southampton and – within five minutes of the opener – Fraser Forster made two huge saves to keep the score down.
The Saints stopper was unbeatable in their recent victory against the Gunners and he made a sensational save – not too dissimilar to Nick Pope's Gordon Banks stop last weekend – to keep out Rodriguez in the 25 minute.
Weghorst looked destined to nod home the rebound but he saw his his instinctive header ricochet off the post from three-yards.
Burnley should have been miles ahead and, just after the half-hour mark, they were nearly made to pay when Romeu – again unmarked from a Ward-Prowse corner – steered a header wide with the goal at his mercy.
Five minutes later, Burnley roared back and the impressive Roberts teed up McNeil, who forced another smart save out of Forster with a low 20-yard drive.
Eventually, the hosts did get their second goal as Collins rose the highest to head Brownhill's out-swinging corner into the far corner. Jack Cork did look as though he was distracting Forster – standing in an offside position – but VAR adjudged that he wasn't influencing the play.
Burnley racked up six shots in the first half – more than they had managed in 90 minutes in each of their last 19 league matches.
And they started the second-half on the same vein that they ended the first: on the front foot.
James Tarkowski headed over from close range after latching onto Brownhill's corner just a minute after the restart.
Southampton did try to rally but they couldn't really threaten the Burnley goal and the hosts thought they had nabbed a third on 64-minutes.
Forster failed to deal with Tarkowski's header across and Cork bundled in the rebound. It would have been another unlikely goal-getter but VAR saw that the 32-year-old was narrowly offside.
At the other end – with literally his first touch since coming on as a substitute – Che Adams had a glorious chance to half the arrears but his volley from ten-yards was pushed over by Pope.
Pope was also equal to a trademark free-kick from Ward-Prowse before Taylor made a sensational recovering challenge to deny Adams a shot on goal.
In the 88th-minute, fans chanted and applauded out of respect for Sean Dyche. Irrespective of how this season ends up, the 50-year-old will never not be a legend on these parts.
But those who said that Burnley would crumble without Dyche at the helm could prove to be very, very wrong. This was up there with their best displays of the season and performances like this couldn't come at a better time.