Willian was inches away from doubling Chelsea's lead early in the second half after Hazard's clever back-heel gave the Brazilian space to shoot just wide. And London-born Rice, who supported the Blues as a boy, took extra satisfaction from frustrating Antonio Conte's men on Sunday. Here's how we rated each Blues player in the game at Stamford Bridge.
A hamstring injury picked up in training has kept him sidelined for club and country since then, but he is now ready to step back into the fold.
Nick Pope was equally impressive in helping Burnley to victory by the same margin against Watford at Vicarage Road, so here Hart needed a response to remind the on-watching England assistant Steve Holland of his qualities.
"We must be realistic", he said.
"They hung on in, battled away and Javier Hernandez came on and did what he's good at".
Chelsea started the game on the front foot and were wasteful at a number of instances before Cesar Azpilicueta put them ahead in the 36th minute. Eden Hazard had the first decent effort of the game as his strike from just outside the box flashed inches wide.
Derby days are understandably somewhat unpredictable, but these are matches that Chelsea can't afford to lose. Chelsea and West Ham are on opposite sides of the capital and the Premier League table, but they were even on Sunday.
On an emotional day where the west Londoners paid tribute to the late Ray Wilkins, they threatened on numerous occasions before Cesar Azpilicueta opened the scoring on 36 minutes.
Rice rivalled goalkeeper Joe Hart for best West Ham player on show. He'll mature with games -he'll get better.
But Javier Hernandez stepped off the bench to drag the Irons back on level terms with 17 minutes remaining and Joe Hart made a fine save to deny substitute Olivier Giroud a late winner.
On forwards potentially lacking confidence: "Morata scored two offside goals".
21st minute PASS: Nelson has been shifted infield by Wenger to give Bellerin a channel down the right. They shout my identify and sing my songs. "Chelsea can get a goal in other ways but from a corner was really bad from our point of view". We have more big teams to come and we will need our keeper to play well.
But it was a disappointing conclusion to a miserable week for Chelsea, who were mourning the death of their former star Ray Wilkins. Draws against teams at the bottom of the table won't help in qualifying for the Champions League.
The Mexico international added: "These types of games are very difficult, you have to be focused to stop a very good side like Chelsea". The club should be more worried of dropping farther than fifth.