It was a first home win since the opening day of the season for the Gunners and, after the game, manager Arsène Wenger said of the victory: "We had a strong team pressure and we had to intensify that in the second half, which we did.
"Slowly you could see that the chances were coming and at the end of the day it became easy when it was 2-0.
"They did very hard and we did make mistakes at the back. Overall I think it was a very satisfying win. Recently at home we haven't got the results we wanted but for the confidence of the team it's also important to win without conceding a goal.
"In the recent games at home we were not patient and we have learned. We look a bit more balanced between offence and defence. We were less in trouble when we lost the ball today, we were serious and that is very important in the efficiency."
Wenger described his star man Alexis Sanchez as a street fighter. "The header he scored is that," he said. "It's more the result of determination and desire than doing something special technically. He has the attributes of a fighter and of a quality, talented player. That's what you want to see in every football player.
"Since he's played through the middle he looks more goal dangerous. Now it looks to me that he's really dangerous through the centre of the pitch because he's in full confidence, he's physically strong, he likes to take people on as well and he makes perfect runs. Through the centre of the pitch maybe he's more dangerous."
Sean Dyche saw his side beaten by a top club and he said: "I feel like the reality of the Premier League is unforgiving and a harsh reality. Teams like this have unbelievable quality and look at the players coming off the bench, let alone who is already on the pitch.
"People might say we have been defensive but we have played two strikers in every single game this season bar one. We want to be brave, but you can't be brave and stupid and I don't think we are. But some teams have quality everywhere and that costs multi-millions of pounds and we don't have that.
"The players are in a high state of learning, including myself, because it doesn't get higher than this. We are ready to try and attempt to compete in this market. We're not ready for the top ten; we know that, but we are ready to challenge beyond that in our real market to see whether we can compete against the bottom teams.
"The players have to grow quickly and I have my fair share of that. I am in a results industry, but there has to be a future to what you do. You can't rely on a lucky win, or another one, because you are never going to improve and build a team relying on luck.
"You have to build it by method and I think ours is reasonably should, but we are coming up against teams who are of a higher level. There is no negative in that.
"There has to be a reality in what we are achieving and looking to achieve. It's been an amazing journey, but there is a different weapon out there that is against us this season."
Dyche added: "I thought the tactical plan worked reasonably well. There was a good energy and shape to the team and pockets of good quality, but the facts are that these teams get ahead and make life very tough. They get you by the throat and don't let go and that's what happened today.
"Tom didn't have too many big saved to make and then, all of a sudden, had four or five out of nowhere. They were on the front foot after getting their noses in front and the second one so quickly was a killer blow.
"The challenge and the reality is not today, or Chelsea or Manchester United. We have played six of the top ten no wand that's not our reality. There are no guarantees there, because look at the signings Hull (who are at Burnley on Saturday) have made. But all the talk at the minute is about getting that first win, but guess what, then you need another, and another.
"The win will come, that's not really a concern. It's when, and then how many you get after that. That's the thing."