West Ham will count themselves unlucky not to have won after having three goals disallowed in the second half, but the victory means they now stand seven points clear of the bottom three, while Stoke are five points adrift of the safety zone.
The Austrian joined West Ham last summer and has been their standout player in their quest to fight relegation, which sees them now six points clear of Southampton in 18th. "I thought we played a really good game and if we keep playing like that, we will win games", Lambert told a post-match news conference.
"The ideal situation is to go unbeaten between now and the end of the season, there is no reason why we can't do that".
Stoke was close to a huge win in its fight to avoid the drop when Carroll, who had been on the field as a substitute for just four minutes, met a left-wing cross with a crisp volley into the bottom corner.
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Arnautovic has been in good form of late too, scoring nine goals and making four in his past 14 games, including setting up Javier Hernandez's equaliser in their 1-1 draw with Chelsea last weekend.
"You always think 40 points is safe".
West Ham ended an eight-game winless streak against Stoke by winning the reverse fixture 3-0 in December.
Coverage of today's Premier League game between West Ham and Stoke City will be shown live on Sky Sports Main Event (Sky channel 401, Sky HD 435, Virgin channel 511) and Sky Sports Premier League (Sky channel 402, Sky HD 436, Virgin channel 512).
The Hammers are aiming to complete their first league double over Stoke since the Potters' first season in the Premier League in 2008-09. "I'm pleased with the team's performance, I thought we played well".
They are one loss short of reaching 1,000 defeats in the top flight and would become the 16th club to reach this landmark. However, his goal had taken West Ham one more step away from danger.