Carlsen vs Inarkiev Illegal move??

It depends. According to the regulation, since Carlsen moved after the initial illegal move, he can't claim the win due to that. Normally when an arbiter sees an illegal move they can immediately claim the loss but in this instance it doesn't appear the arbiter saw it at the time.

I'm not sure why Inarkiev stopped the clocks and called the arbiter over. Carlsen did not make an illegal move. He was checked, he moved his king to a legal square. Claims of illegal moves do not have to be made. If Carlsen had taken the king, instead of stopping the clock to claim an illegal move, that itself would have been illegal and could have resulted in a game loss, from my understanding of the regulations.

But he ended up getting the win since the chief arbiter overruled the initial arbiter's decision and forced the game to be continued (I'm guessing with Inarkiev to move and get out of check) and Inarkiev refused to continue.

Inarkiev made the faulty assumption that Carlsen played an illegal move by missing the win claim. This is wrong according to FIDE blitz rules and after the chief arbiter read the rules loud (with Norwegian TV covering all the drama live) to him, he still refused to accept it.

We can never know for sure if it was an intentional dirty trick, but he stopped the clock the second after Carlsen made his move and claimed the win, looked very suspicious. Also they mentioned on TV that other players claimed that Inarkiev has done similar things in blitz earlier. Seems like poor sportsmanship to me, and completely unexpected at this level.

To the OP question: It's not a win for Carlsen since he missed it and made his move, Inarkiev will then get another chance to move out of check. 

Btw:

By the updated FIDE rules from 1 January 2018 an illegal move will no longer lead to direct loss, but rather additional time for the opponent.

Interesting. That's going to confuse some people

Lol yes, they like to toss the rules around every now and then. When I played my first OTB blitz tournament many years ago an accepted way to claim the win was capturing the king like you mentioned above, later this has been changed so that reply is an illegal move now. 

I believe these changes might lead to some losses amongst club players, but I am amazed that professional players don't know the rules well, like we saw today.

And here is the FIDE ruling on Inarkievs complaint:  https://www.fide.com/images/stories/NEWS_2017/FIDE_News/Inarkiev_Claim.pdf

I think in OTB blitz and bullet in USCF rules you are allowed to just zap the king and win if the opponent doesn't see the check....???? I have been at a OTB tournament and seen it about 10 years ago. Is FIDE different?

Yes they are. It's one of the big differences.

I'll have to look at the rulebook again but taking the king in USCF isn't allowed as I recall. They had matched the FIDE rule of stop the clock and claim the win. As of January 1 of 2018 (for events starting then or after), it appears to just be a time penalty of 1 minute in blitz, 2 in rapid and a loss on the 2nd illegal move for FIDE.

 I thought it was a lot cooler and funnier too when you could just zap the King.....I will never forget the look on this smarta.... kid's face when that happened to him.

That was in earlier times the take of the king and win. Some guy tried this to me but me was told that I could claim a win because to take of the king was illegal too. So he had to claim before me instead of take the king of.  Now it is thus again changed in you never lose after an illegal move. So good to read and to remember! I think it is better by the way to recontruct the position just for the illegal move happened add extra time to the opponent and continue the game.

I must say that the rule if an illegal move is not claimed by the arbiter nor the player the game shall continue. I did not know this rule and I must say because in this example the game can not be continue because the black king is in a check. So this rule is rather strange to me. I am glad that it is changed.  How can a game continue after the king on b6 is in a check ? ( you have to go out a check according to the rules. )  So Ne3 was illegal  Kd3 was also impossible. But Carlsen did not take the king. What happens after such a thing I also did not know.  The most logical thing to me seems if you notice this incorectness you recontruct the position to the move that everything was ok so to Rxb7+  and then Ne3+ must be replaced by an other move that is legal. I hope they will change the rules to this. And because of the Ne3 illegal move you can add time to Carlsen. (  To play on with a king in check sounds really ridiculous to me.  Imagine how long they can play on then for serveral moves with one ( or both ) kings in check? Before one notice. Isn t that weird?  For me unacceptable and undesirable. )

Reconstructing the board position right before the illegal move should be the best option instead of continuing with illegal move. I really don't understand of the logic of Inarkiev claiming a win for his own illegal move!

The blitz rules are clear on this point, if the opponent fails to see / claim a win on illegal move and performs his own move, the illegal move shall not be reverted. The game should continue from where Inarkiev made a faulty claim and stopped the clock. Reading the actual rules are much better than a lot of people arguing here about what they think makes more sense...