Sean Dyche vowed to steer the Clarets out of choppy waters, following a third straight defeat.
The Clarets twice came from behind at the London Stadium before finally losing the game 4-2 through a brace from the hosts in the final six minutes.
And Dyche insisted he and his players will emerge fighting from the tough spell of recent results.
He said: “We’ve had lots of plaudits over the last six years and this is probably our first really tricky spell, so I’m not going to cry it in.
“We have work to do, without a shadow of a doubt, but I couldn’t be more proud of the players and what they have all done to get where we are.
“You can’t just have the wheel of a ship in calm waters. You have to take it when it’s choppy, and you learn a lot more from that.
“I am still learning as a manager and I think the players are because it’s the first time they have had a few things go against them, so there’s work to be done but I think the mentality on show today was very positive.
“They didn’t get their rewards today, but if we keep going about our business in that manner we will get our rewards.”
The Hammers opened the scoring early in the game as Arnautovic capitalised on a mistake from James Tarkowski, who will visit a specialist this week on a hernia problem.
Johann Berg Gudmundsson equalised on the stroke of half time, only for the Londoners to regain the lead through Anderson.
Substitute Chris Wood again levelled matters with a powerful header 13 minutes from time, but with both sides pushing for the winner the hosts snatched it through Anderson’s second on 84 minutes and Hernandez’s late clincher.
Dyche added: “It was certainly a ding-dong of a game, especially for the neutral, but not so when you are on the wrong side of it.
“I’m pleased with the mentality of the players, that’s for sure, because we gifted them the first and reacted to that - and by the end of the half we were forging our way into the game.
“We go 2-1 down and then get it back to 2-2, then their third goal kind up sums up what’s happening right now and is a real sickener because Tarky gets a block and it falls to their man, then the ball hits Ben (Mee) and it goes in the wrong side of goal.
“We react again and Woody hits the bar, but then at the end we are chasing it when they get a breakaway.
“We certainly don’t want to be conceding the number of goals we are, but I was pleased with the mentality to keep going at them to try and get something from the game.
“I’ve always mentioned the margins in games and, as I said, I’m not going to cry it in now, because last year we were on the right side many times.
“Some of what’s happening we can sort out, and sometimes things just go against you.
“The nature of the side is coming back. However, we know that we have to keep the ball better and be tighter than that.”