The Dragons boss was under pressure before Friday night's Hampden meeting, having guided them to just one win from their first four Group A matches. But the 2-1 success for Coleman's men in Glasgow moves the Welsh up to third place, where they are seven points behind joint leaders Belgium and Croatia.
Wales meet Croatia at the Liberty Stadium on Tuesday, and captain Williams told Press Association Sport: "The manager was spot on against Scotland. He made good substitutions and got all the big calls right. There has been criticism of him but not from anyone who matters. People have just been talking for talking's sake."
Williams added: "We are not concerned about it and the lads are behind him. We enjoy playing for him, we enjoy the camps, we enjoy the training. Against Scotland you saw that he knows what he is doing."
The fact Wales' victory - which they came from behind to achieve - came without the assistance of 25-goal Tottenham forward Gareth Bale - who limped off at the break due to ankle problems and a virus - was another satisfying fact for Swansea defender Williams.
He said: "It was a very important win for us. We need the points and there was a little bit extra riding on the game because it was against one of the Home Nations. It was massive. But we showed we can cope without Gareth. He dug in at Hampden and gave it a go but you could see he wasn't himself. He tried for 45 minutes but it wasn't right.
"But after that we came through and no one can say we are a one-man team. We were a goal down and came out for the second half without him. But we got the result and everyone played their part."
For much of the first half in Glasgow, Wales' passing and movement was a joy to behold. The hosts, however, were able to eventually gain a foothold in midfield which they used to steady themselves before Grant Hanley's first intentional goal.
Wales established their command and Williams now wants to see his side boss proceedings for the full 90 minutes when the Croatians visit, saying: "That's the best I've seen from a Wales team. That is us at our best. That is how we want to play. We kept the ball well and didn't let Scotland settle on it. We won it back early and that is what we are supposed to do.
"We dropped off a bit towards the end of the half and they scored but in the second we controlled it pretty well. If we can play like we did for the first 25 minutes for a full game, we'll be a match for anyone."