Defender Harry Darling had given the Swans the lead briefly heading home Josh Tymon’s 36th-minute cross, before the hosts would respond immediately through Siriki Dembele to leave the two sides level at the break.
Jamaican international Jamal Lowe’s sixth goal of the season restored the away side’s lead after the break with Tymon once again the supplier after neat build-up from Matt Grimes and Liam Cullen had made space for the former Stoke City wing-back to cross.
The Swans would come close to extending their advantage with substitute Jerry Yates agonisingly close to turning in Jamie Paterson’s ball across the face of the goal.
It would be heartbreak for Williams though on his league bow in charge of the club as Welsh international midfielder James would smash an unstoppable effort past Carl Rushworth deep in stoppage-time to share the spoils.
It would be a satisfactory afternoon for Williams with progress being made, but the Londoner feels improvements need to be implemented.

“We showed a toughness I liked because we came under quite a lot of pressure at times,” said Williams.
“We do though need to start to play cleaner. There are too many scruffy actions, so I have to help the guys make those repetitions and try and understand the ideas better.
“If we do that hopefully we can control the game more, I don’t think either team was particularly able to control the bulk of the game.
“We expected an energetic performance from Birmingham because Tony Mowbray always makes sure that happens.
“We caused ourselves problems because with a team really fired up that press you and be aggressive, you can’t afford to play sloppily, especially near your own goal.
“We created a few good chances, Birmingham created a few good chances too. There were some encouraging things, there was some quality on display and some great character. Those are things you can really work with.
“It was disappointing [too conceded so late] because there was such little time but if you take the game overall, Birmingham deserved a point.
“I think we made the classic mistake of trying to defend the lead out.
“I need to encourage the players to keep playing football in the same way, because if it’s good enough to get us in the lead, it’s good enough to stay in front.
“However, we didn’t create enough clear-cut chances to have won. Maybe three good chances, but you can’t win a game like that if the other team also creates the same.”
[Lead image Swansea City FC]
