Caparros was dismissed after five games in charge, even though his side were unbeaten in the last four and had climbed off the bottom of the league.
"It's been an intense month's work and a more than interesting professional and life experience," the former Athletic Bilbao coach said on his website.
His departure came one week after fans displayed banners in support of the Caparros, who led the Spanish side to sixth in La Liga last season, during a Swiss League match.
Chagaev took over the small club in the lakeside town in mid-May, saying he wanted to take the club into the Champions League, and coach Didier Olle-Nicole was fired on the day of his arrival.
He was replaced by Bernard Challendes, who saved Neuchatel from relegation in their final three league games of last season but was fired after they lost the Cup final 2-0 to Sion.
Brazilian Sonny Anderson was then hired but moved to general manager because he did not have a coaching licence and Francois Ciccolini was appointed.
However, Ciccolini was dismissed after two games of the new season along with Anderson and the entire coaching staff.
Chagaev has also parted with all the club's sponsors and fired the administrative staff since his arrival, leaving the club unable to print tickets for their opening match of the season.
In another extraordinary move, the club fired goalkeeper Rodrigo Galatto after one game and one week at the club.
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