City were not at their best before the break as Bukayo Saka's penalty cancelled out Kevin de Bruyne's opener.
But they took control in the second half with Jack Grealish and Erling Haaland both on target to send City top of the league for the first time since August on goal difference.
"Very happy to get the three points, that's it. Very intense, many duels, two teams who love to press high and have the ball. Fortunately, we were the ones on top at the end," Dias said after City's 11th successive league win over Arsenal.
"This means everything to us, when you come to these moments against your rival, it just means everything. For our team that's the way to live it. One game at a time, but leave it all there in that one game and make sure you have nothing left."
Norwegian goal machine Haaland had been enduring a relatively barren spell having not scored for three games.
But he produced a marauding second-half display and sealed the win with his 26th league goal of the season in the 82nd minute, 10 minutes after Grealish had fired City ahead.
"A small adjustment in halftime from Pep and in the end we have good quality players and we have to get it out of every player and we did today," Haaland said.
"We can all agree (Arsenal) have been the best team this season so to come and play against them is not easy, but we played an amazing game and got three really important points -- we are in it again."
Haaland laughed off ending his mini-drought.
"It has gone 20 minutes since I last got a goal --so I have to keep working," he said.
"It was so positive, happy celebrations, and I am so happy with everything, we needed this one and now we have to go on a run, because that's what Manchester City should do."
