Maatsen slotted home in the 18th minute to give his side their first win after their six-year spell in the top flight came to an end last season.
Burnley's line-up underwent drastic changes during the close season as several players left to remain in a top-flight league, including winger Dwight McNeil and defenders Ben Mee and Nathan Collins who stayed in the Premier League.
"I keep saying, on any given day, give me hard work and attitude and I'll take that above talent and that's what Burnley is," Kompany told Sky Sports after the match.
"We're not there, don't get me wrong, but it's one step in the right direction, showing what we could do."
Former Manchester City skipper Kompany joined the club in June after leaving Belgian side Anderlecht, who sealed European qualification after a third-place finish and reached the Cup final under his lead.
