Goals from Juninho Bacuna, Lyle Taylor and Onel Hernandez brought the Blues' recent struggles to an end, and similarly halted the Hatters' excellent run of form in 2022, which had taken them up to seventh after Tuesday's win over Barnsley although Wednesday's results had seen them drop back down to ninth.
Manager Jones made one change from the side that won in midweek, with Tom Lockyer coming into defence in place of Gabe Osho, who was named amongst the substitutes.
Cameron Jerome was back on the bench against his former club, whilst Harry Cornick and Elijah Adebayo were partnered up-front for the second match in a row.
Town started well on top, with the opening quarter-of-an-hour played almost exclusively in the Birmingham half until the home fans brought the match to a halt by throwing tennis balls onto the pitch from the stands, in protest against the St Andrew's club's owners.
The only opening of note created came in the 12th minute, when Campbell stood a cross up from the right byline, aimed at Adebayo, whose flick landed at Amari'i Bell's feet, but the ex-Blues left-back couldn't get enough purchase on his effort to really trouble keeper Neil Etheridge.
Town seemed to be knocked out their stride by the hold-up, especially when a steward came several yards onto the pitch to pick up a solitary tennis ball just as Adebayo looked to fashion an opportunity to cross on the left edge of the penalty area.
Referee Darren Bond stopped the game again, awarding a drop-ball, and from that point on the early momentum stalled.
It was the hosts who took a 25th-minute lead with their first real foray in the Town box, Gary Gardner crossing from the right and Lyle Taylor reacting first to the loose ball at the near post, with Burke and Lockyer around him, and laying it back for Bacuna to rifle into the roof of the net,
The Hatters mounted an immediate response, Adebayo flicking on Shea's long kick forward into the path of Bell, but the wideman's cross-shot was held by Etheridge when a cut-back towards Cornick at the far post might have been a better option.
Just 15 seconds into the second half the Blues' lead was doubled, Lansbury inadvertently prodding a Jeremie Bela ball infield past Burke, who was coming out of the back line to intercept, to present Taylor with a clean run on goal, slipping his finish low past his former AFC Wimbledon team-mate Shea.
Kal Naismith attempted to force a brerakthrough for the Hatters in the 56th minute, driving forward into the Birmingham half and unleashing a low left-footed drive that whistled just past Etheridge's left post.
Jones made a double substitution a minute later, bringing on Fred Onyedinma and Carlos Mendes Gomes in place of Lansbury and Lockyer.
It was almost three in the 64th minute, when Jordan Graham's free-kick was spilled by Shea, under pressure from Kristian Pedersen, but Adebayo got back to deny Bacuna a second of the afternoon by clearing his weak effort off the line.
Tje respite was brief, however, and it was three by the 69th minute, however, when Graham's cross from the right flicked off Burke's head in the six-yard box and Hernandez was lurking at the far post to finish past Shea.
There was to be no way back for the Hatters, who saw substitute Osho booked for a late foul on Bela, with attentions now turning to next Saturday's visit to Kenilworth Road of West Brom, who moved back above tenth-placed Town on goal difference.
Goals:
BC - Bacuna 25, Taylor 46, Hernandez 69
