Howe, whose first two matches have been Championship away fixtures at Scunthorpe and Portsmouth, received a rousing reception when he was introduced to the Burnley fans before kick-off on Saturday.
Chris Eagles' double and a late goal from substitute Martin Paterson on his return from injury prolonged Howe's honeymoon period at Turf Moor.
But not before Calvin Zola had marked his Burton debut with a goal to make it 2-1 and set up a tense finale.
Howe, who as Bournemouth boss last season completed a league double over Burton, named the same starting XI that won at Portsmouth on Tuesday night.
This was Burton's first game for three weeks since their giant-killing performance against Middlesbrough in the previous round.
Boss Paul Peschisolido made just one change from that 2-1 win at the Pirelli Stadium, with Zola coming into the side following the departure of leading scorer Shaun Harrad to League Two rivals Northampton.
The two teams were separated by 59 places in the Football League prior to kick-off and it was evident in a largely one-sided first half.
Adam Legzdins was almost embarrassed when he fumbled Danny Fox's free-kick in the sixth minute but the Burton keeper managed to grab the ball before it went over the line.
Eagles then went close with another free-kick soon afterwards.
Jacques Maghoma shot straight at Lee Grant in a rare Burton attack, but Burnley were soon on the attack again and Steven Thompson shot just wide after 14 minutes. Jay Rodriguez then headed wide from Eagles' cross.
Burton almost went ahead against the run of play when Scott Malone's terrific free-kick clipped the top of the crossbar with Grant beaten all ends up.
Eagles made Burnley's pressure count in the 29th minute when he collected Jack Cork's backheel and weaved his way into the box before sliding the ball beyond Legzdins.
Rodriguez got a run on goal but dwelled and the chance went begging, before Thompson went close with an angled shot moments before the break.
Burton were better in the second half, restricting Burnley much more and creating a chance of their own when Maghoma threatened with a 20-yard shot.
It was not until Clarke Carlisle headed over from Fox's cross just before the hour mark that Burnley looked dangerous again.
Legzdins kept his side in the game when substitute Ross Wallace cut inside from the right wing and shot from the edge of the area, the Burton keeper pushed the effort away for a corner.
However, two minutes later Eagles gave Burnley a two-goal cushion, pouncing on a dreadful defensive error.
After Legzdins' goalkick had struck Darren Moore on the back of the head, Wallace scampered down the right and crossed for fellow substitute Chris Iwelumo who set up Eagles for a simple tap-in.
Paterson came on for Wade Elliott in the 73rd minute, ending a two-and-a-half-month injury absence, and he almost headed in a cross from Eagles shortly after his introduction.
Eagles was unlucky not to complete a hat-trick in the 75th minute when his 18-yard shot struck the crossbar and bounced down on to the line.
Wallace should have put the game beyond Burton soon afterwards but skied his shot after a cutback from Paterson.
And it was a mistake by Wallace, surrendering possession in midfield, in the 80th minute that saw Zola set up a tense finale with a neat flick past Grant from six yards out from Malone's cross.
However, Paterson settled any nerves with a third goal in injury time after Legzdins had spilled Wallace's free-kick.