Brazil midfielder Guimarães struck in the second half as West Ham’s claims for a foul in the build-up were ignored.
Eddie Howe’s side are up to sixth place in the Premier League, level on points with fifth-placed Manchester City and two points behind fourth-placed Chelsea.
A top four berth will guarantee Newcastle return to the Champions League after missing out this season.
Even finishing fifth could be enough to ensure qualification for Europe’s elite club competition depending on the results of the remaining English clubs in continental action.
After losing three of their last four matches, the Magpies’ restorative victory was perfectly timed ahead of their second League Cup final appearance in three seasons.
Newcastle, beaten by Manchester United in the League Cup showpiece in 2023, face Premier League leaders Liverpool in the final at Wembley on Sunday.
The Tyneside club are aiming to win their first major trophy since the 1969 Inter-Cities Fairs Cup and their first domestic prize since the 1955 FA Cup.
“One of our best wins of the season, not in terms of performance but desire and heart,” Howe said.
“I felt we were in need of a win to go into the final with a real boost. It will do us the world of good,” he added. “Also for our league table position today was vital.”
West Ham boss Graham Potter added: “We didn’t do enough to win the game but we could have got something out of it.”
“The goal was disappointing, we felt there was a push on Kilman,” he said. “It’s two hands on the back. You can see from Isak’s non celebration he thinks it’s a push. Clear for us but I’m not VAR.” (AFP)
