Wales were second best for much of the 1-1 draw in the Baku sunshine, with Switzerland enjoying 64 per cent possession and having double the number of efforts on goal – 18 to nine.
But Kieffer Moore’s excellent 74th-minute header cancelled out Breel Embolo’s effort just after half-time to earn Wales a point in their Group A opener.
“You go one-nil down and you can really crumble,” Wales captain Bale told BBC Sport.
“We showed our class to keep playing and fight for the goal back. It was a good equaliser.
“We worked hard and it was difficult in the heat as well.
“We showed a lot of fight, determination and character to come back from one-nil down and I’m proud of the boys.
“You don’t lose so you keep going. We have to use this as a springboard for the next game.”
Interim manager Robert Page had played a false nine in Wales’ warm-up friendlies and left Cardiff striker Moore on the bench.
But Moore was handed his first competitive start during Page’s tenure and rewarded that call with his sixth goal in 18 appearances for Wales.
The result will place greater emphasis on Wales’ next game against Turkey – who lost their opening fixture 3-0 to Italy – when they clash in Baku on Wednesday.
Page said: “I am not disappointed by the performance.
“I thought we were brave on the ball, with Joe Allen dropping back into the penalty spot and playing through was excellent.
“That togetherness and grit the players showed was no surprise to me, I see it day in and day out.
“I’ve said to them that’s the foundations where we go now in this competition.
“Every game we want to go in to win it and it will be no different on Wednesday.
“If four points is enough (to qualify for the last 16), brilliant. If not we’ll keep going because we don’t fear anybody.
“We respect teams in this competition and if we have to get four points then we’ll be ready and go again.”
Switzerland next play Italy in Rome on Wednesday before returning to Baku for a final group game with Turkey.
Substitute Mario Gavranovic thought he had volleyed home a winner six minutes from time as Switzerland recovered from Moore’s equaliser to finish strongly.
But Gavranovic’s effort was ruled out for offside following a VAR check.
Switzerland head coach Vladimir Petkovic said: “We didn’t take advantage of the opportunities we had and we have to be better in the next two matches.
“If we create opportunities in those games we have to be more clinical.
“We were a little bit more passive after they scored, but Wales had to take risks and put in a stronger performance after falling behind.”
On the group situation, Petkovic added: “We have known from the start that Italy are the clear favourites in this group.
“But we haven’t lost any gap to the other teams.
“We are still in this group and we will try to get points against Italy.”