Arnautovic will be hopeful of delivering another decisive performance on Monday against his old club, who are six points from safety with five matches remaining.
"We've still got four games so no-one is safe".
"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".
Hart's failure to impress came while his Stoke counterpart Jack Butland - another contender for the England number one shirt - produced a solid display. However, the West Ham boss believed his side should have wrapped the game up long before the late drama.
The former Liverpool striker's first goal since November put relegation-threatened Stoke within touching distance of their first win in 10 games.
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. I thought Javier Hernandez and Manuel Lanzini helped, all the players have played well.
The Hammers are aiming to complete their first league double over Stoke since the Potters' first season in the Premier League in 2008-09. Obviously he hasn't played for 11 weeks so I have got that on my mind as well.
West Ham have only won back-to-back Premier League games at the London Stadium on one previous occasion back in December 2016.
"I think if the team had been playing this way from August, I wouldn't be here and they wouldn't be in the position they are in".
Although Hart was relieved his howler didn't cost West Ham all three points, his mistake did little to boost his chances of starting for England in the World Cup.