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It's Match of the Day at 22:30 BST on BBC One.
Gary Lineker is joined by Alan Shearer and Danny Murphy.
Leicester City manager Claude Puel speaking to BBC Sport: "I don’t know if the result is fair but we take the three points and it’s an important cushion for us.
"Brighton played well and they had quality so it was difficult for us in the first half after the international break. In the second half it was better and the penalty save from Kasper changed the situation. It gave us the belief to score both goals.
"Of course the save is a crucial moment and it changes our mentality and our desire and perhaps it was the opposite for our opponent."
Asked whether Wilfred Ndidi's sending off was fair, Puel said: "I don’t understand why he was sent off because when I watched the reply it wasn’t a foul. He touched the ball and it changed direction.
"I hope it can be watched by the commissioner again because he is a crucial player for us.
"It was important to show that desire this afternoon and we have to give it everything every time we play. Today the first half was not enough so it will be important for us to correct this in the next game."
Brighton & Hove Albion
Brighton and Hove Albion manager Chris Hughton speaking to BBC Sport: "We are hugely disappointed and a game we didn’t deserve to lose. I thought the things we needed to do to nullify their strengths we did very well, but the next bit is scoring goals ourselves.
Asked whether the penalty miss changed the balance of the game, Hughton said: "They are the moments in a game we want to go our way but it doesn’t and that gives them the advantage.
"It happens and Glenn (Murray) has scored penalties and vital goals for us this season. Leicester will be grateful for a good save and we are disappointed we missed it."
On the final run-in: "Every game feels like that last push we have got very difficult games coming up and next up is Huddersfield at home.
"There were a lot of aspects from today that we will settle for in that game but we need goals because that will get us points. Any nerves come with the territory and by playing against top sides in the Premier League but we will need to be on top of our game next week."
Burnley striker Ashley Barnes speaking to MOTD:
On their performance: They cause a few problems in first half, we had to stick together. We got that goal and it killed it off a lot of their momentum. In the second half we were a little sloppy to let them back in.
On his goal: Best goal so far for sure I’ve scored. As I checked back, I just swung my foot at it and it went in. I’ll probably watch that a few times tonight.
On personal ambitions: I set out a target, still way off the target for myself, but main is to help the team as long as we keep picking up points, who know.
How far can Burnley go? Our main aim was to finish on more points than last year, we just have to keep building on that and see how may points we can get.Copyright: Rex Features
West Bromwich Albion
West Brom manager Alan Pardew:
On his side's performance: It was a difficult afternoon up against a good side. Second half was a lot better, I tried to change the shape and it worked better. We were unlucky, after having a chance at 2-1 not got get something out the game in the end.
We were so tentative in trying to close the ball down, that was not the team I see in training. we didn't cope with the pressure well, but we got better in the second half. We need to get a result somehow, and did everything today.
On battling relegation: You learn that certain players can deal with the situation, others not. We have another tough match next week, we have to be confident enough to to stand up to the opposition and ask them questions.
Leicester City midfielder Vicente Iborra: "We are very happy.
"Today the win is very important for the team as we begin the last period of the season.
"We won the game with some good work after Kasper’s save. After that save the team started to believe and we scored two goals.
"This is a nice moment for me because I want to help the team. The win is the most important thing for us, but I will help with goals where I can.
"I think we have to be ambitious and work hard every day but we are playing in the correct way and next we play at home. It is important to win in front of our fans."
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Brighton & Hove Albion
Brighton defender Shane Duffy: "Obviously we are disappointed after having the chance to go ahead in the game but they caught us on the break and it killed us."
Asked about the penalty, Duffy said: "It's just defining moments in the game.
"(Glenn) Murray has been great for us all season and you fancy him at that point but it wasn’t to be.
"We felt comfortable and that we were the ones going to win the game but they defended well and it’s a setback for us.
"We have been here before and we know we can bounce back and get a run going. We've got a huge game against Huddersfield next week and hopefully we can get the place bouncing.
"We expect it to go down to the wire and the teams around us are all picking up points so it was disappointing to lose at home today."
West Ham United
West Ham captain Mark Noble: "I knew the fans would get behind us today. We have been in tough situations before and I knew they would turn up and make noise. I could tell from the moment I walked out of the tunnel that they were fully behind us.
"Collectively as a team and a set of fans we came together today and got what we deserved.
"We lost to Burnley with everything that went on but we could have been 2-0 up in the first half. Today we took those chances.
"It's a lot of pressure on us out there and we dealt with it tremendously. When Marko plays like that he is unplayable."
More from Mark Hughes: "My relationship with Marko Arnautovic when I worked with him at Stoke was very good, you'll have to ask him about that [goal celebration]. I don't know if he was pointing at me or was just excited by scoring."Copyright: Reuters
Burnley manager Sean Dyche:
On the result: Really pleased, another big important win. We want to be as strong as we can. It is a strange place to come being how tough a time they're having, but thought we handled that well.
On the performance: We scored a really good opening goal. We had more possession than you would expect for an away side, we were effective with it, and opened it up with a fine goal.
On potentially breaking the top six: Not sure about that. there is progress being made, that's the marker. It's hard to make massive strides. Can we keep building? Can we keep moving forward? And we're doing that.
On qualifying for Europe: You have to be careful what we wish for, we're a work in progress. To be the real deal, got to keep working, but I am proud of my players' achievements.
Southampton manager Mark Hughes: "We were poor. The opening 10 minutes of any game is key, you have to manage it correctly and we damaged ourselves. We conceded poor goals and gave us too much to do.
"We have to find a way to score goals. If you can't get a foothold in the game by working hard and winning the right to play you are not going to create chances. That is the focus this week. We have to understand that we are in a situation of our own marking.
"We have to be better next time out. The games ahead are going to test us greatly, we have to be ready and we have to have answers.
"From my point of view it was a good learning experience. It's OK training well but you have to be able to translate that onto the pitch on a Saturday afternoon. Today we couldn't do that."
"[Arnautovic gestures?] I wasn't aware of that - Marko played exeptionally well today as he did in four-and-a-half years with me at Stoke."
Huddersfield boss David Wagner to BBC Sport: "It's a disappointing result for us but it's my job to analyse this in the right way. Our idea was to frustrate them and sit deeper to minimise their clearcut chances and hurt them on the counter attack.
"In the first half Newcastle had good opportunities but in the second half I thought it was working for us. We had a big opportunity with Scott Malone at the final whistle, we didn't use the chances and this is why we lost the game.
"The attitude, fighting spirit and how they followed the game plan was fantastic.
"I think we were worth a point."
West Ham United
West Ham manager David Moyes to MOTD: "We had a great start, great support from the first minute and it inspired the players.
"We think we're good enough to be clear of the bottom three. Recent results haven't been good but we played well.
"The supporters were great, they have been great here bar a few. The players gave them something to shout about today.
"We think Marko Arnautovic is doing a great job for us, he could easily have had a hat-trick today. He's making a big difference to us.
"Over the time you come up against people, Marko has had words with Mark Hughes in their time together but it's nothing worth talking about."
Swansea forward Tammy Abraham on Manchester United keeper David de Gea: "He showed why he's the best keeper in the league. I was surprised he saved my two shots. I'll stay positive and still try and take my chances.
"It's a final push now and we have to keep believing. We'll go into the next game all guns blazing and hopefully we'll get the result."
More from Jose Mourinho on his team's season: "Last season we won titles but finished sixth in the league.
"We want to finish second this year and have 10 points more than last season, more goals scored, less goals conceded, but there is a club which is making it practically impossible to follow. In other leagues we would be fighting for the title.
"We are having a positive season and we still have the FA Cup to come. We are going to fight for our presence there."Copyright: Reuters
West Ham United
West Ham boss David Moyes: "The players were great, their prep over the last few weeks, their attitude has been great, they wanted to put past events behind them.
"The supporters were great and they appreciated the performace of the players. We got the job done early which helped.
"They have worked together over the years [Arnautovic and Hughes], Marko is that type. People who have worked with him know that, they will tell you. I'm seeing that now, I enjoy working with him and he is showing people why West Ham paid £20m for him from Stoke."