The visitors, meanwhile, are searching for their first victory of the season after playing out a goalless draw against Birmingham City in their season opener.
Kompany may have taken over a side that were low on confidence following their relegation from the Premier League, but they looked like a rejuvenated force in the opening game against Huddersfield Town.
A first-half strike from Chelsea loanee Ian Maatsen proved to be the winner, with Burnley preventing their opponents from registering a shot on target.
While they were resilient in defence, the Clarets also dominated with the ball, enjoying nearly 70% possession to a template of the style of football that Kompany expects from his players.
The Burnley manager will hope that his side can implement their possession-based game once again on Saturday in Kompany's first competitive game at Turf Moor.
Facing a side they have not competed against since 2007, Burnley will try to extend their unbeaten run in meetings with Luton to five matches.
Meanwhile, Luton are hoping to enjoy another successful campaign after surprising many pundits by securing a playoff place last term.
A 2-1 semi-final defeat by Huddersfield Town ultimately ended Luton's Premier League dream, leaving the Hatters to prepare for a fourth consecutive season in the Championship.
Nathan Jones has responded to the playoff defeat by making nine additions to his squad over the summer in a bid to replicate last season's success.
A few of those summer additions were included in the starting lineup for the season opener, where Luton had to settle for a point following a goalless draw against Birmingham City.
After drawing last weekend, Luton know that they will have to beat Burnley for the first time since March 2000 if they are to claim their first win of the season on Saturday.
After missing the season opener, Jay Rodriguez has returned to training this week, but Kompany may not risk the forward for Saturday's contest.
With the hosts looking to begin the season with consecutive victories, Kompany may be tempted to opt for an unchanged lineup for his first competitive home game as Burnley boss.
Burnley possible starting lineup:
Muric; Roberts, Harwood-Bellis, Taylor, Maatsen; Costelloe, Bastien, Cork, Brownhill, Cullen; Barnes
Luton Town possible starting lineup:
Horvath; Osho, Bradley, Potts; Bree, Campbell, Cornick, Clark, Bell; Adebayo, Morris
We say: Burnley 2-0 Luton Town
Burnley impressed in their first game of the season, and we think that Kompany's side will have enough quality to beat Luton on Saturday, which would ensure the Clarets begin the campaign with consecutive victories.
For data analysis of the most likely results, scorelines and more for this match please click here.
body check tags ::
Previews by email