With four minutes left in the game, Arnautovic nearly won it for West Ham as he raced towards goal but was stopped by a superb tackle by N'Golo Kante.
Saints remain three points adrift of safety despite taking the lead early on through a Shane Long goal and then leveling at 2-2 when Charlie Austin scored just seconds after coming on as a substitute.
West Ham went into the game on the back of a morale-boosting 3-0 win over fellow strugglers Southampton last weekend, to ease rising tensions between the board and supporters.
Morata thought he had notched up the second, but after latching onto a misplaced ball by West Ham, Morata raced goal wards, passed two his left to in-running Willian. he cross the ball along the six-yard box and Morata skilfully ticked home.
"We can create a lot of chances but we cannot take them".
The Blues dominated the match at Stamford Bridge but only had Cesar Azpilicueta's first-half goal to show for it. "Chicha has come on - after being on the other side of the world and being poorly - and we're so glad he was fit and got that vital goal", said Hart.
"Chelsea were really good in the first half but we kept it to one".
In real time, it looked as though the Chelsea man had one the ball, but as replays so often do, doubts were raised after the game as to whether the ball had been won legally. To say that Chelsea's results in the past few months have been disappointing would be a colossal understatement.
Many feel Liverpool have their top-four place pretty much sealed at this stage in the season, the lack of reaction to the Reds dropping points against Everton on Saturday underlines this.
Blues fans will probably not get the chance to watch their side perform in the Champions League next season, and will have to be content with the Europa League.
Unsurprisingly the game got off to something of a subdued start, although Eden Hazard sprang into life in the fourth minute and nearly gave Chelsea the flawless start.
Conte believes the draw describes Chelsea's season perfectly, given how many chances they squandered.