Martin Odegaard struck a 50th-minute winner at the Emirates Stadium on Sunday after Morgan Rogers had given Chelsea an early lead.
Kai Havertz equalised in the first half for Arsenal, who have a 100 per cent record after three games and moved level at the top of the standings with Manchester City.
"Our mentality is to keep fighting. We have shown that many times," Arsenal skipper Odegaard told Sky Sports.
"We never want to concede early but the reaction is top. We got our reward in the end and the response was brilliant. We showed our quality with and without the ball and the result reflects that."
Manager Mikel Arteta added: "Very disappointed with the way we conceded the goal, but it was a joy to watch our team play, compete, some huge individual performances as well and the atmosphere in the stadium was great to witness."
Earlier, Manchester United 's tough start to the season continued with a 2-2 draw at Everton, despite twice leading at Hill Dickinson Stadium.
Ainsley Maitland-Niles struck in added time to rescue a 2-2 draw for Everton against United.
The transfer deadline day signing came off the bench to make his debut and curled a brilliant long-range shot into the top corner to salvage a point for his new club.
United twice led through goals from Bryan Mbeumo and Benjamin Sesko, who looked to have scored the winner in the 88th minute.
But Everton had an answer on both occasions, with Tyrique George also scoring with a long-range screamer.
"Everybody wants to start off like that," Maitland-Niles said.
"The ball fell to me on the edge of the box. I put my foot through it, it ended up in the top corner, which I'm happy about."
It was another setback for United, which has just one win from their opening three games of the season after a loss to Hull and victory against Ipswich.
Defensive frailties cost United again as they conceded two goals for the third game running.
Mbeumo fired United in front straight after halftime with a low shot into the far corner. George lashed in a near-post equaliser in the 83rd, but substitute Sesko headed United back in front.
The visitors were seconds away from three points when the ball broke to Maitland-Niles outside the box in the sixth minute of added time and his first-time shot left United goalkeeper Senne Lammens standing.
"We haven't started the season that bad. It's not that bad, we've got that close to coming away with three points today, that's football," United coach Michael Carrick said.
"All the work that went into putting us in that position at different stages to then come away like that – it's not a great feeling. It feels like a defeat."
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