And it looked as though the Irons were going to make it seven points from a possible nine as they controlled much of the game against Paul Lambert's side.
Sobhi was proving Stoke's primary creative threat and he sent Mame Biram Diouf clear with a cute first-time pass around Declan Rice, but the Senegal striker fired straight at Hart.
The offside flag came to the visitors' rescue again on 66 minutes when Edimilson Fernandes beat Butland with a low shot from outside the box, but Arnautovic was in the gloveman's line of sight.
Andy Carroll's volley salvaged a point for Moyes' men but Hart will be in the spotlight after a second mistake in four games since being restored to the West Ham team.
The only noteworthy chance for West Ham until that point had fallen to Arnautovic, inside the six-yard area, with a snap shot that struck Jack Butland in the face and flew out for a corner.
Stoke have lost their last four league matches to sit second from bottom with 27 points and Masuaku believes his team can take advantage of their rivals' struggle for consistency.
Away from home the Potters haven't tasted success since October 2017, failing to win any of their last 12 matches on the road in all competitions.
All the same, it was easy to see why they were averaging under a goal per game this season.
"If they'd done that at the start of the season there'd be no way they'd be in this position, absolutely no way".
Clearly vice-chairman Karren Brady didn't get the memo about just how important Monday's game was, though, judging by her PR blunder 15 minutes before the game.