The Bewtway Premiership side got off to a great start when Lucas Ribeiro broke through the Dortmund defence to fire past Gregor Kobel after just 11 minutes.
But an error by Williams gifted Felix Nmecha an equaliser five minutes later.
Teboho Mokoena then lost possession in midfield and a quick break forward ultimately saw Serhou Guirassy put Dortmund ahead just past the half hour mark, with a towering back-post header.
Jobe Bellingham further extended the German club's lead with his first goal in his first start for his new team, on the stroke of half-time with a composed finish.
Khuliso Mudau was, meanwhile, unfortunate to deflect a 50th minute cross past Williams and into his own net to seemingly put the result beyond doubt.
But the Brazilians fought back and were on the scoresheet again when Iqraam Rayners expertly headed home his own rebounded effort from a teasing Mokoena delivery, as a late Lebo Mothiba goal ensured a tense final few minutes for Nico Kovac's charges.
Speaking after the game, the Sundowns skipper explained that at this level, every mistake is pounced upon.
He said on IOL: "Of course, it's not nice to concede a goal like that but we are a team that takes risks.
"Before I received the ball, I saw Tebza [Teboho Mokoena] there but he got a call from Marcelo [Allende] to move away, so it was just one of those unfortunate situations but we learn from it and we keep going.
"At this level, if you make mistakes, they punish you and we learned the hard way today. I think all of their goals were probably mistakes, errors from us and at this level, you get punished. Where we come from, we get away with it. This is a good lesson; we shouldn't be disappointed."
Williams added though that the gap between the two teams isn't as big as he expected.
"We showed that we can compete. Yes, obviously they [Dortmund] are physically bigger but it's all about your mind...if your mentality is right, you can go out there and compete against the best and that's what we showed today," he said.
"I am just proud of the boys for the effort, the desire and the fight. The gap isn't that big and I think we proved that once again today."
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