Newcastle actually created the better chances at a wet and windy Emirates Stadium but were made to pay for their continued wastefulness.
Aleksandar Mitrovic and Georginio Wijnaldum came closest to scoring for the visitors, who came up against an inspired Petr Cech in the Arsenal goal.
But it was Koscielny, scoring his third goal of the campaign, who ensured the points stayed in north London and Walcott feels Arsene Wenger's side have now matured into one capable of winning ugly when required.
"The three points are crucial at this moment in time, especially when teams slip up as well," he said.
"We've had a blip at times but it's important that we get those hard-fought three points today. In previous seasons we would have lost that or drawn, so that's a very good sign for 2016.
"The title is anyone's to be honest but there's such a belief in this team that we can do it this year. It's been a long time and everyone understands that, but you could just sense with that performance that we know what we need to do.
"It's up for grabs but we have the strength in depth in this squad and I believe that we can push on this season and hopefully be successful.
"A lot of games have been like that this season. The Premier League is starting to open up a little bit and a lot of teams are ready for the battle this year.
"The performance wasn't there to be honest, but the work rate is always there and when you have work rate you always get the success that you need, want and deserve."
Newcastle have now drawn a blank in their last three matches, losing all three by a tight 1-0 scoreline as they remain in the Premier League bottom three.
Steve McClaren's side have now taken one point from their last four outings after back-to-back victories over Liverpool and Tottenham gave them renewed hope of survival.
Wijnaldum may have passed up one of the better chances of the game at Arsenal but insists the players will keep going as they look to end their goalscoring hoodoo.
"I thought we were the better side," he said.
"We had good precision play and created chances but we didn't take them, so it was going to be difficult. They had one set-piece and scored.
"We had a lot of confidence that we could play football here, but in the end we have nothing and we're sad - it's really frustrating. You know they have the quality to score but if we had taken two chances we would have won.
"You must keep faith that you are going to score goals, but every time you don't you get frustrated and disappointed. But we have to keep our heads up and we must fight to get the points."