The Isthmian premier team’s effort against their National South hosts was hampered by a penalty and a red card given against them in the first half-hour.
Barnes felt the 11th-minute spot-kick and the yellow cards that led to defender Sam Okoye being sent off – both in the first 30 minutes – were harsh.
That left United with a mountain to climb, but after going 3-0 down midway through the second half, they battled back with late goals by Femi Akinwande and Tyler Christian-Law and might even have nicked a 3-3 draw and forced a Pilot Field replay.
The exit leaves the Us to focus on their blossoming Isthmian campaign and Sussex Senior Cup.
They host winless Potters Bar on Saturday and visit Lewes on Tuesday in a busy week of league action and will host Three Bridges in the county cup on Tuesday, October 3.
Barnes praised the support Hastings had at Dover and said: “I’m not one to make excuses but the officials had too much of a bearing on things early on.
"It was a soft penalty and after getting an early booking, Sam got a second yellow for alleged time-wasting. It spoiled the game.
"In the end we made a real fight of it. We had a good chance that we missed to make it 3-3.
“It’s a shame to go out but the league is of course our No1 priority and I’d have swapped a cup run for three points.”
That’s the prize Barnes now wants from each of the next two games, which he said would be two different types of test.
Tomorrow’s visitors Potters Bar are one off the bottom while Tuesday’s hosts Lewes have had a good start and are only three points behind Hastings.
Barnes will be without banned Okoye tomorrow but hopes to have Adam Lovatt back after injury.
"We want to be in among it and many say you can tell a lot by looking at the table after ten games, which we’re approaching,” he said.
"People will expect us to win Saturday but it will be a hard game, while Tuesday will give us a real measure of where we are.”