Agutter, who joined the club in 2017 initially as assistant manager, went on to be appointed first team manager following the departure of Adam Hinshelwood.
Saturday's 3-1 defeat to Felixstowe & Walton United in the FA Trophy proved to be Agutter's last game in charge.
Hastings United CEO and chairman Billy Wood said: "Chris approached me Sunday for a meeting and when we met Sunday night he offered his immediate resignation from the role with the feeling he wants a new challenge, a challenge he doesn’t think the job at Hastings United offers anymore.
"Whilst shocked and saddened by his decision the club respects his position and have stood him down from his management duties.
"Chris has been a fantastic manager in his time in the role and I am proud of our record over the past three seasons.
"Most importantly the club comes first and we will make sure the staff are supported this week with two league fixtures.
"The goals of the club do not change and the belief in our squad remains the same. Tuesday night let’s get right behind our team like each and every one of them deserve."
In the interim, assistant manager Andrew Brown will take over managerial duties and will lead the team tomorrow night against Whitstable Town at The TGS Pilot Field.