Emile Smith Rowe capitalised on a Chelsea mistake to score the only goal of the game, and extend our winning run in the league to three games.
Having scored his first ever Premier League goal at the weekend against West Brom, he netted again in the derby to set us on the way to a hard-earned three points.
This goal owed a bit more to good fortune though, as he stroked home from Pierre-Emerick Aubameyang's pass, once Chelsea keeper Kepa Arrizabalaga had scrambled back to prevent an own goal.
For much of the rest of the game it was about resolute defending, right from the front back to our five-man defence.
That was the system Mikel Arteta went for from the start, but it was the most central of those three centre backs - Pablo Mari - who was caught in possession early on to gift Chelsea the first chance of the game. Kai Havertz nicked the ball off the Spaniards toes to race away on goal, but he blazed over when one-on-one.
It was Chelsea's turn to hand out the gifts next though, and it led to Smith Rowe scoring the only goal of the game.
Jorginho's wayward backpass put Kepa in difficulty. The keeper managed to palm away from his own goalline, but Aubameyang was lurking to pull the loose ball back to Smith Rowe in front of goal, and his low shot squirmed in off the post.
Rob Holding made a superb close-range block from Mason Mount's shot to keep us ahead, but Chelsea kept knocking on the door for the remainder of the half, drawing two smart stops from Bernd Leno.
The hosts continued to pin us back at the start of the second half, and we had to remain organised in and around our own penalty box.
Christian Pulisic did have the ball in the net, but the goal was disallowed for a clear offside after VAR intervened.
Hector Bellerin was introduced and immediately had a sight of goal, but shot tamely after good build up play involving Martin Odegaard.
We were certainly coming out of our shells as an attacking threat now, with Odegaard himself blazing over from a promising position.
Chelsea - three days before their FA Cup final appointment with Leicester City - piled forward late on and hit the woodwork twice in succession in injury time.
But we were rewarded for an excellent defensive effort with a 1-0 win, securing our first double over Chelsea since 2003/04.
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