Javier Hernandez believes the club are moving in the right direction but has stressed the importance of retaining momentum after Sunday's result, with The Hammers unable to afford many slip-ups.
A quick pass from Arnautovic found Hernandez and the striker, on for just three minutes, drilled a fine finish past Thibaut Courtois.Marcos Alonso nearly spared Chelsea's blushes with a thunderous strike, but Hart finished as West Ham's hero with superb saves to keep out that effort and one from an Olivier Giroud header.
This had been another lacklustre performance, but Joe Hart took two strides to his left to produce a superb one-handed save on to the post to preserve West Ham's point that surely ends the Blues' hopes of Champions League football next season.
Still, it was not all bad news for the beleaguered Senegal star.
The victory stretches Arsenal's winning streak in all competitions to six successive matches and lifts them to 54 points, three behind Chelsea, who in turn trail Tottenham in fourth by ten points.
In terms of importance West Ham got the better of it as their point puts them six points clear of the drop zone, Chelsea remain fifth in the Premier League.
Hart was needed, though, to salvage a point for the visitors as he made brilliant stops from Marcos Alonso and Olivier Giroud in the closing stages to do his hopes of starting for England at the World Cup no harm.
Shane Long gave them the lead after 17 minutes, nipping in ahead of defender Shkodran Mustafi.
But with just nine minutes left on the clock Welbeck grabbed the victor. The talented 19-year-old has become a key man in the Hammers first team and he also made his senior global debut, against Turkey last month. None of these will be easy games for the Hammers. And in a still-tight Premier League relegation battle, every point can be crucial.
Irish worldwide Long's goal was only his second in nearly 50 games for club and country but even before that the visitors made all the chances.
'Some gamers are injured and coming again.
Welbeck then grabbed his first in seven league games when his effort from outside the box was deflected past goalkeeper Alex McCarthy via the unfortunate Maya Yoshida's boot.
It was no more than Southampton deserved when substitute Austin marked his entrance by tapping in Cedric's low cross. But Welbeck had the last word by heading in the victor moments after he had missed a much easier chance.
Blues defender Antonio Rudiger told the club website: 'The thing is we have to start playing good football and winning games, because if we don't win games we can not get in the top four.