Just eight minutes later Noble was shown a straight red card for a high, lunging tackle on Wilfred Ndidi.
West Ham frustrated Leicester, while Lukasz Fabianski made some crucial second-half saves.
Jamie Vardy had dominated the pre-match discussion, with reports the former England striker was angry at being dropped for the game.
The 31-year-old, who was forced off against Arsenal on Monday with a stomach bug, started the game on the bench and Leicester manager Claude Puel said the player was not “100% fit”, despite Vardy declaring himself so via Instagram on Friday.
As it was, Vardy came on at half-time and offered more of an attacking threat. He went close with a header and late in the game had a fierce strike saved by Fabianski.
With the Hammers poised to take all three points, Ndidi’s long-range shot took a wicked deflection off Balbuena to beat Fabianski.
There was concern at the end of the game, as Leicester right-back Daniel Amartey left the pitch on a stretcher in injury-time in some pain with an injury to his ankle.
Puel addressed the Vardy rumours at the end of the game, he said: ” I don’t know why it is a problem.
“I wanted him to come on in the second half to win the game. I have to manage his physical being. It is difficult to play him when he isn’t 100%.”
Puel celebrated a year in charge at the King Power this week, but left the pitch on Saturday night to boos.
The Frenchman had gone into the game on the back of four defeats in his last six league games and his team sheet raised eyebrows before kick-off, making four changes from the team that at lost 3-1 at Arsenal on Monday.
Selecting Kelechi Iheanacho instead of Vardy was the most eye-catching choice and the Nigerian barely threatened in the game, apart from when he was flagged for offside in the second minute, having swept the ball into the back of the net.
Vicente Iborra had the Foxes best chance in the first half after Noble’s red, but his header was straight at Fabianski.
Vardy’s second-half introduction gave his side more control, sharpness and pace in attack, but still Leicester toiled.
They had a flurry of chances, 21 shots in total, with 15 coming in the second half, but many were either from long range or offside – they were flagged seven times in the match.
Harry Maguire hit the crossbar with a header on 67 minutes and Fabianski kept out Marc Albrighton twice.
Puel introduced Shinji Okazaki for Turkish centre-back Caglar Soyuncu with 10 minutes remaining and finally got the breakthrough via Ndidi.
After getting back into the game, Leicester almost threw it away minutes later as Angelo Ogbonna shot over from a few yards out.
“I was happy with the second half because it was a good mentality and there was quality on the pitch,” said Puel. “We showed two faces in our game and of course, we need to resolve this.”
