Burnley claimed a first win at the Emirates Stadium, and first at Arsenal since September 1974, courtesy of Pierre-Emerick Aubameyang's headed own goal, as the Gunners finished with 10 men after the dismissal of Granit Xhaka, and were lucky not to have Mohamed Elneny sent off.
It was a second win in four games for the Clarets, who climbed out of the bottom three as a result, and Dyche said: "We've had a tough start - we thought it might be because of the injuries, but we've responded well to the challenge in front of us, which is apparent, you have to start winning games.
"The energy and belief has always been good, and it's going back to where we were, not a massive surprise because we are getting people fit again.
"When that fitness comes, you get a bit more depth and that competitive instinct, a group fighting for their shirt, and you take on the games in a better shape.
"We still have people coming back, and it doesn't guarantee you anything, but it increases your chances of performing, and if you're performing, you have an increased chance of winning games."
Asked whether a big win would give the players confidence, Dyche added: "I don't think we've lacked confidence, I think inner depth of confidence improves when you get results of course, and it knocks away a few stories and question marks.
"We've still got lots of work to do, but I think there have been more positive signs over the last four, five, six games maybe.
"We certainly had a golden chance first half and created some other moments, we still know we have to tidy up in the front third, but the performance was built on what we've done many times, that kind of strength of unity, willingness to work for each other, and finding the right moments and margins.
"I think we did that well tonight.
"They're still a good outfit, there's a lot of noise around then but they still have quality players, and I think we used that well to almost put them out of the rhythm they are looking for, and make it awkward."
Dyche felt Arsenal should have finished with nine men: "You know what I'm like with the physical side of the game, I think it's to a level which is absurd now, you can't touch anyone, but the rules are the rules, and if Xhaka is getting sent off, I don't know how Elneny doesn't.
"VAR and all that...I'm a big fan of VAR, but that takes five seconds, as soon as I saw it afterwards, that gets done straight away for me.
"But it didn't.
"Credit to them, they responded well, the 10 men situation gave them something to almost grow again, that freedom comes weirdly because the expectation changes, and I thought they responded well, it almost became a cause to take on.
"The next 10-12 minutes they had a couple of chances, Popey made saves you'd think he would, but he still has to make them.
"Then we calmed it down and looked a little more in control, without being in total control because they're still a good outfit.
"We nick a good goal with good delivery and saw the game out pretty well after that."