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Mohamed Salah's 23rd minute strike separates the two teams. It appeared like the Seagulls would get steamrollered in the moments after, but they held on and also managed to cause palpitations in the Liverpool defence, particularly Alisson in the Reds goal, who got away with one.
And here's Hughes on Hojbjerg's dismissal for diving: "In my opinion, going down to 10 men was unfortunate. To defend Pierre, I think both yellows were a bit soft. The first one was just a coming together. But if you've got one, you walk a tightrope. I think he was clipped when he went down on the second, but Mr Moss was in position and he sent him off."
Southampton manager Mark Hughes: "I thought we were excellent first half. On the front foot, asking questions of the opposition.
"We got our noses in front with a fantastic goal from Ryan but if we managed that game better, we could've come away with something. At that point, I thought the only winner was us, but momentum shifted and that was a really hammer blow to lose at the end.
"The manner of the performance gives me hope for the rest of the season because we definitely deserved to get something out of today's game. We let them back into the game and it's really cruel that we haven't got anything to show for it. It was a really strong performance and if we continue like that, everyone will be happy."
Everton boss Marco Silva: "It's not the best way to start. It's something you can't do even if it is minimal contact. It's something he can't do, him or other players in our squad.
"We showed fantastic attitude with the 10 players. The first 25 mins of the second half we did a fantastic job. At 2-0 we cannot concede from counter attacks and you have to improve in those moments because it's very important to keep the three points.
On Michael Keane's head injury: "He went to the hospital. I wait for updates. It is not a good thing for us. I hope it's nothing special."
West Ham United
More from West Ham boss Manuel Pellegrini, speaking to Match of the Day: "They won 3-1, but if you analyse the game you see we had so many options to score the goal.
"I am very happy in the way we played, we played well with good pace and movement, so I am sure soon it will come.
"The team played well, it was a team that defended well. I am happy as a team we are improving on a lot of things we are trying to do.
"We will see during the week what happened with Marko (Arnautovic) - he felt a pain in his (knee) muscle. (Michail) Antonio has a pain in his ankle."
Bournemouth goalscorer Joshua King: "It was crazy. The boys showed character and being 2-0 down we managed to come back and grind a point out.
"Last season we got a lot of points by coming from behind. You don't want to chase the game but it's the character in the group."
The Mo Salah song echoes around Anfield after the "Egyptian King" puts the Reds ahead with a first-time sidefoot finish.
As it stands, Liverpool will go top of the Premier League.
Arsenal manager Unai Emery, speaking to BBC Sport:
"We are happy because it is the first three points after two defeats. It was very important to win, and we showed in the match that in difficult periods we can win, after 1-0 in the first half. We suffered with the transition in the first half, but we improved in the second half and controlled more like we want. In the second half we wanted to find more balance on the pitch and we did.
"After the substitutes, they made the difference in the second half. Today I am happy and for the confidence it is good, for the consistency it is good also.
"We need to improve. To improve in this transition to get the balance, to get more control with the ball but also, West Ham are a good team, it was a difficult match, for that we are happy but there are things to improve."
On Mesut Ozil's absence from the squad:
"Yeah he was ill and yesterday we decided with the doctor and with him that he would not play. Today he is here and with the team also because he feels better today but he is not here now because he was sick yesterday."
Huddersfield boss David Wagner, speaking to BBC Sport: "I think from my point of view the first hour we controlled the game. There was one team who wanted to do something in the offence and create opportunities and this was my team. We have known that in the last 10-15 minutes if we keep the tempo high we will get the space but unfortunately this was not the case because we went one-man down.
"In the end this was a good point for us, the lads fought. The incident, to be totally honest, there were two aggressive players one was more clever than the other player and unfortunately it was not my player who was clever.
"They had two clear cut chances, we defended well, as good as we good, and we still tried to make the lucky punch but we could not find it. It was a good point for us."
West Ham United
West Ham manager Manuel Pellegrini: "We created a lot of chances and I was very happy in the way we played, we dominated parts of the game and were unlucky with the own goal.
"We improved on what we did in our two first games. We need to add points, but we are going to continue searching.
Unfortunately we had two injuries with (Marko) Arnautovic and (Michail) Antonio playing very well. The style, we need to continue doing with more trust than ever after losing this game.
"In every game we are improving."
Bournemouth boss Eddie Howe: "I didn't see Everton's sending off at all. I heard the crowd reaction and that was all.
"The best spell of the game for us was when we went down to 10 men against 10 men. Adam Smth's red card was very harsh - it was the slightest of touches.
"It's another point gained. It's been a great start for us. We're looking forward to Chelsea next."
A "harsh" red card? Watch the incident on MOTD at 22:30 BST, BBC One.
BBC Radio 5 live
Huddersfield midfielder Philip Billing: "We had chances to score. Against two big centre backs every time we crossed it was probably breakfast for them. We just need that goal but it will come, for sure.
"We go into every game to win, this game we thought we had a chance to win; The red card changed the game. I thought we showed great character, I am comfortable we will be OK if we play like this."
Leicester City manager Claude Puel: "Tough game. Southampton played very well in the first half with intensity and it was difficult to find space. After, we found more space in the second half and i think the result was deserved.
"We dominated, we had a lot of chances and we had two or three chances more to score. First half it was good defending and I think we deserved this because we could've scored before second half."
On Harry Maguire's match-winner: "Harry is a very good player for us. It is fantastic for him to score, for him and the team, it was a very good game. We have to keep this mentality to the end when we come back in the game. it's fantastic."
On Hojbjerg's dismissal for diving: "I didn't see the replay but if it is simulation, it is a natural yellow card."
More reaction to come. Next up...
West Ham United
West Ham goalkeeper Lukasz Fabianski, speaking to Match of the Day: "It was pretty busy for me, but you expect that against teams like Arsenal away from home.
"The most important thing is the team and we lost, so I cannot be fully happy. We had some pretty good chances, a good chance at the end of the first half to go 2-1 up.
"Their second goal was a little lucky. Some details can cost you the game, but we can be pleased with the performance. With a bit more work and belief in what we are doing we can improve and do better.
"There were periods of the game where we looked solid, compact and like a team that wanted to go forward and attack.
"It's all about keeping belief, because when you look at our squad there is a lot of quality individually.
"There is still a lot of work to be done, but from what I have seen today we are going in the right direction."
Arsenal manager Unai Emery, speaking to Sky Sports: "The three points were very important. Today I saw good things, but also things for improve.
"It was very important for our supporters. After 1-0, the team was improving, the players, the substitute players made the difference. It was for that I am happy but there are things we must improve.
"More transition. For the second half we were speaking in the dressing room and to help more inside with two midfielders. We need the balance, in the first half we suffered more than we want.
"It is clear for us we need to improve. Also, the win gives the confidence for the players. The first half, we needed calm. The players, they show every supporter we want and can do the second half going forward with balance, and win the match.
"We need this control with possession with the ball and also going forward with balance, which is important."