Crystal Palace goalkeeper Wayne Hennessey was the man of the match with a series of fine saves from Miralem Pjanic, Vedad Ibisevic and Haris Medunjanin in Friday's Euro 2016 Group B qualifier in the Welsh capital.
But the hosts - who won their opening group game 2-1 in Andorra last month - also had chances of their own, with Williams heading over from close range and Gareth Bale testing Asmir Begovic with a stunning stoppage-time strike.
"It's a good point," said Williams on Sky Sports.
"We feel that we played well and it would have been nice to get the three points, but as a team I think we showed how much we've grown.
"It was a battle. We knew it would be as they are a good team and we said before the game we had to stand up and be men - everyone who played has really dug in."
Manager Chris Coleman echoed those thoughts and also paid tribute to the home fans, who gave their side outstanding support from start to finish of a hard-fought encounter.
"I think the performance pleased me even more than the point and the atmosphere was incredible, the supporters were magnificent," commented Coleman.
"It's the proudest I've been as manager and I thought all the lads were heroes out there. We said before that if anyone wants to take anything off us here, they'll have to work hard."
Williams was left to rue his missed opportunity as he failed to control a header after a wicked free-kick cross from Bale.
"I can't believe it," he added.
"I just got a little bit in front of it and had to try and come back, but I should have scored. I'll be kicking myself all night about that."
However, Hennessey can have no regrets about his stellar performance as he held firm under intense pressure at several points in the match.
"All the lads put in a great performance, not just me," he declared. "We always knew Bosnia have got great strength but these boys were fantastic."
Wales now host Cyprus on Monday with four points from two Group B games, while Bosnia-Herzegovina face Belgium after getting off the mark with this stalemate.
