Chris Coleman's side sit top of Group B after David Cotterill and Hal Robson-Kanu goals helped them to a win in Cardiff.
Wales were 2-0 up inside 23 minutes, however Cyprus pulled one back before the interval through Vincent Laban - and the hosts ended up playing most of the second half with 10 men when Andy King was sent off.
Bale hailed Wales' hot start and said the mood in the camp was all positive - after they went unbeaten in their two qualifiers during the international break, having drawn 0-0 with Bosnia-Herzegovina on Friday.
"It was a great start from us again. We came out of the blocks firing, like we wanted to," Bale told Sky Sports.
"We got a good start to the game, obviously a little mistake off a set-piece, but I think you can see how much the lads want it.
"It's a different feeling in the camp now, everyone's really fighting for each other, the team spirit's amazing and the fans again were literally unbelievable."
Bale believes Wales' endeavour to continue to run with 10 men showed how desperate they are to reach a first major championship finals since the 1958 World Cup.
"I think you need a bit of luck sometimes. Obviously it didn't go for us today [with the dismissal], but I think we showed hard work," the Real Madrid star said.
"Since the sending off, we worked hard as a team. I think - again - we covered every blade of grass and showed how much we want it."
Bale said the international break was a successful one for Chris Coleman's men, giving them plenty of confidence for their trip to Belgium in November.
"It's obviously a great place to be. I think the minimum from these two games was four points, we knew it was going to be a difficult game against Bosnia and a winnable one tonight," he said.
"I think when guys like [Fulham youngster George Williams] come in, they do a fantastic job. It's a big game against Belgium, and we'll go there full of confidence."
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