The Cherries won 2-0, with Arsenal having William Saliba sent off on the half-hour mark.
Cook later said, "Massive. They are a top, top team, the football they have played in the last few years. But the lads dug deep, controlled the game in big spells and got the win.
"We did a lot of work in the week. They have a lot of attacking players that can change the game. The red card helped so we could take control of the game.
"Sometimes when a team go a man down they can start playing well, like they did at (Manchester) City. But we created the chances needed to win the game. We know we want to be clinical as each goal in this league is massive.
"Arsenal are so compact. You think of them as an attacking team but they can adapt. We matched them today and got the win."