Diafra Sakho's late goal condemned Sunderland to a 1-0 defeat that left them one point above third from bottom Burnley.
It was hard on the visitors who were the more enterprising side before the break and deserved a point.
With an international break next week, Advocaat has two weeks to prepare for a crunch North East derby at home to Newcastle United.
"We have to win that game," the much-travelled Dutchman told Sky Sports.
"With our home fans, if they stand behind us, like they did today because they appreciated what we did, we will make it very difficult for Newcastle. This game has given us the tools."
Sunderland striker Jermain Defoe missed his side's best opportunity when he lashed a shot over the crossbar in the first half after being teed up by Steven Fletcher.
"It was a very difficult game but out team did really well working-wise and fighting-wise because West Ham are a very difficult team to play against because they only play long ball," said Advocaat who has been coach of Netherlands, Belgium, South Korea, Russia and Serbia.
"We had the first chance in the first half with Jermain Defoe. With his quality, that was a 100 percent chance.
"He had to at least hit the goal. We can take that with us into the next game because if you work hard you get the results too. I have seen some things we need to improve and some players are coming back. So for me it is positive."
(Reporting by Martyn Herman, editing by Tony Jimenez)
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