The Bahrain-Merida Belgian covered the 13km test against the clock in 17 minutes and 57 seconds, with the race jury giving Fuglsang the same time.
Haig finished 24 seconds off the pace, putting him 59 seconds in deficit to the Dane on the general classification but leapfrogged notoriously poor time trialler Mikel Landa.
The Australian's impressive second overall surprised even himself.
“It is a great feeling to move up...especially going into the TT in third place and not really expecting to move up," Haig said.
“If you would have told me I would come away with second place before the race then I would have been incredibly happy."
"Even with the little mishap the other day in the final, it has still been an incredible week and the guys have done an amazing job helping me everyday.
"For the new guys in the team Alex Konychev and Andrey Zeits, it was their first stage race with us and I think it worked really well.”
Haig's Mitchelton-Scott team mate Alex Edmondson sat in the hot seat for most of the day until Teuns bettered his time by two seconds, towards the end of the stage.
Their compatriot Chris Harper (Jumbo-Visma) also impressed with eighth place on the stage (31 seconds behind Teuns/Fuglsang) and 14th overall.
“This is a good result. I felt really good today”, Harper said.
“It was a very tough and lumpy time trial. It was going up and down a lot and we even had a bit of offroad in it as well.
"The plan was to ride uphill as fast as possible and then recover a little in the descents. I was able to keep my pace. It also confirmed that my shape is good and that, despite my mechanical problems of yesterday, I can look back on a very good week.
"Things don’t always go the way you want. But the legs felt great all week. On the climbs it went well and my time trial was okay too. This gives me a lot of confidence for the future.”