Ten Hag was tasked with bedding in more than a dozen new signings before this season, having lost several key players in the summer.
Florian Wirtz joined Liverpool for a fee rising to £116m, Jeremie Frimpong also moved to the Reds for £29.5m and Granit Xhaka signed for Sunderland for £13m.
Defender Jonathan Tah joined Bayern Munich on a free transfer, winger Amine Adli moved to Bournemouth in a deal worth up to £25.1m, while goalkeeper Lukas Hradecky joined Monaco.
The three most expensive signings in Leverkusen’s history – Malik Tillman from PSV Eindhoven, Jarell Quansah from Liverpool and Eliesse Ben Seghir from Monaco – all arrived this summer, for a total of £88.4m (102m euros) plus add-ons.
“This decision was not an easy one for us,” added Rolfes.
“We firmly believe in the quality of our team and will now do everything we can to take the next steps in our development with a new set-up.”
The club did not name a replacement for Ten Hag, saying the “training work would be taken over by the assistant coaching staff for the time being”.
“We remain committed to achieving our goals for the season – and to do that, we need the best possible conditions at all levels and across the entire first team,” added Carro.
“Now it’s a matter of fully implementing and utilising these conditions again.”
Leverkusen, last season’s Bundesliga runners-up, are in the Champions League this campaign – with Manchester City on 25 November and Newcastle on 10 December among their opponents.
Their first game back after the international break is at home to Eintracht Frankfurt on 12 September (19:30 BST kick-off), a match you can watch live on the BBC Sport website, app & BBC iPlayer.
