Diafra Sakho equalised before half time, but Norwich were back in front on 83 minutes when Nathan Redmond took advantage of more sloppy play from Noble to smash home inside the box.
Cheikhou Kouyate fired home a loose ball in the third minute of injury time, though, to earn the Hammers a point.
Here are five things from a West Ham perspective which we learned from the game...
1) Hammers showed they have fight as well as talent
West Ham have been making the headlines this season thanks to a great start which has seen them beat Arsenal, Liverpool and Manchester City away from home as well as Newcastle United at the Boleyn Ground.
The Hammers scored nine goals and conceded only once in those four wins during which Slaven Bilic's side showed what an attacking and entertaining force they are this season.
But while the London outfit would have fancied themselves to beat newly-promoted Norwich on Saturday, what the game did prove is they have the fight and spirit to never give up when things aren't going to plan.
It proved that Bilic's men have more than one string to their bow, and that bodes well for the rest of the season.
2) The last season at the Boleyn Ground can only get better
The Hammers will be delighted with their start to the season given that they are sitting pretty in third place ahead of a trip to basement boys Sunderland.
However, things have not been perfect in what is their last ever season at the club's beloved Boleyn Ground home of 112 years. Concerns after back-to-back home defeats to Leicester City and Bournemouth at the start of the season were eased somewhat by a comfortable 2-0 over Newcastle.
But the Hammers showed again on Saturday that they are struggling to really get into their groove at the Boleyn Ground, failing to beat a supposedly inferior side.
Away from home the Hammers have been perfect so far with three wins from three in the Premier League, but the Farewell Boleyn season hasn't quite gone to plan and at this rate it will be one to forget rather than cherish when they move to the 54,000 seater Olympic Stadium next summer.
The positive is that there is plenty of time to put things right and lots of big games to look forward to before then.
3) The penny HAS to drop for tough tackling Andy Carroll
Andy Carroll has been told by both Sam Allardyce and now Slaven Bilic that he must alter his approach if he is to steer clear of injury.
The big striker has spent long spells out throughout his career so far and made his first start in eight months last week against Leicester City in the Carling Cup.
Both Big Sam, as reported by the Mirror, and Bilic – quoted by the same paper feel that Carroll must moderate his aggressive approach to avoid needless injuries. His latest long term setback in January came after a robust challenge up at Liverpool and when he tried to play on it a few days later at Southampton it was made worse.
One of the main reasons Hammers fans love Carroll, and have shown such patience with him through all the injuries, is that committed, all-action approach. His style is a throw back to the halcion days of English football, but while it might win him admirers it is also impacting his career.
On Saturday again, with less than 90 minutes under his belt since his comeback, Carroll – on as a second-half substitute – went thundering into a challenge which left him hobbling around for a few minutes. Hammers fans feared the worst, as Bilic probably did too, but luckily the big Geordie seemed to shake it off.
On the other token, his late brave challenge in the air with Noriwch 'keeper John Ruddy set up Kouyate for the equaliser so it is a difficult balance. With his size and frame he must be careful to avoid impact injuries even if it does take away a slight edge to his game but keeps him on the pitch and scoring goals.
4) Adrian is so good he can even save with his face
Goalkeeper Adrian has been one of the stars for West Ham since his return from suspension at City. Apart from strikes which he could do very little about, he has saved almost everything that has come his way.
And he was at it again on Saturday when he stopped a point blank, full-blooded volley from Cameron Jerome... with his face. The Spaniard didn't know too much about it but was in the right place and stood tall as Jerome arrived and unleashed a stinging drive at goal.
Jerome smacked the Hammers number one square on the nose and rebounded to safety leaving him poleaxed in the six-yard box. It did not dampen his spirits, though, as he soon jumped to his feet, lambasted his defenders and eventually saw the funny side.
5) Mark Noble must find consistency if he is to keep his place
When Noble was handed the Hammers captaincy by Bilic after Kevin Nolan's departure earlier this season, it was not a huge surprise. After all, he is Mr West Ham.
But it did raise eyebrows given that many fans felt his place in the team, unlike previous seasons, is no longer a given. Having signed the likes of Dimitri Payet, Pedro Obiang, Victor Moses, Michail Antonio and Alex Song to add to a midfield already containing Kouyate, Noble actually started the season out of form.
It followed on from a very poor end to last season and some fans questioned whether appointing him skipper was a wise decision given he cannot be guatanteed a place.
However, since being handed the armband, Noble's performances have stepped up significantly. That level dipped considerably against Norwich, though, as he was at fault for both the visitors' goals.
While he is undoubtedly a vital member of the squad, Noble will know his performances have to be of a consistently high standard if he is to keep his place.