Scotland took the lead just before half-time with Grant Hanley's header, but the loss of Gareth Bale through injury failed to dishearten Wales and they levelled with Aaron Ramsey's 73rd-minute penalty after Robert Snodgrass' foul earned a second booking.
Wales piled more punishment on the 10-man hosts a minute later as Hal Robson-Kanu's towering header kept their faint qualification hopes alive, but Ramsey must serve a ban after a late red card.
"We had to be patient. We felt for the first 30 minutes we dominate. The players had given everything we asked of them," Coleman told Sky Sports.
"We were patient, we kept popping the ball around. We tried to move Scotland around because they have got some strong boys, but then I think we became over confident and we complicated everything.
"They had one or two chances where they scared us and then they scored at a real bad time for us before half-time. We just said at half-time, go back to what we are doing, keep it simple, keep moving it and keep moving them around.
"Unfortunately we had to bring Gareth off at half-time. He's ok, I think he had a bang on his ankle which was sore at the start of the week. Like we said, no risks."
Crystal Palace midfielder Jonathan Williams replaced Bale at the break and Coleman was full of praise for the teenager after his energetic display helped Wales to launch their fight back.
He said: "Young Jonny Williams is a super little player. To come to this place, a big stage, intimidating atmosphere, 19 years of age, to go on and play with that sort of confidence you know. Superb.
"We just got back to it in the second half. We started popping it off and got our just rewards."
Wales have moved up to third, seven points behind second-placed Croatia, who they host in Cardiff on Tuesday.
"If we'd have lost tonight questions would have been asked of me again," said Coleman.
"We've won. I said we needed six points from two games, we've got three here.
"We've got to pull ourselves back together for the big one down in Cardiff on Tuesday."