“It has not been good enough and we need to do better,” Moshiri wrote in an open letter to Everton supporters.
“Mistakes have been made and for that I want to apologise to all of you.”
Moshiri and chairman Bill Kenwright have faced criticism from fans this season for the way they run the club, which included the unpopular appointment of former Liverpool boss Rafael Benitez last June.
Benitez was sacked in January after a run of one win in 16 games and replaced by Frank Lampard who, after a difficult start, secured their top-flight status for a 69th successive season.
Moshiri says an ongoing strategic review, which has led to the appointments of Lampard and director of football Kevin Thelwell, is evidence of the board’s determination to learn from past mistakes.
The British-Iranian businessman also reaffirmed his commitment to deliver a “fully funded” new stadium at Bramley Moore Dock, which is currently under construction, while pledging to not make the same mistakes in the transfer market that led to the club posting combined losses of £372m over three seasons.