The 2020-21 Formula E champion was drafted in to replace Alex Albon on Saturday at Monza, after the Thai driver was struck down by appendicitis.
De Vries qualified a solid 13th before being promoted to eighth due to a raft of grid penalties being applied to drivers ahead of him.
The Dutchman only lost one place in the race to finish his debut in ninth, keeping that result despite being summoned by the FIA over an erratic driving infringement – for which he was given a reprimand.
Commenting on his manic F1 debut with Williams, de Vries said: "The whole of the last 24 hours have just been a dream.
"I didn't really have much time to think because everything was so rushed. I had a very bad sleep, it went from excitement into nerves and I didn't dare even look into my sleep tracking because basically I spent the whole night awake.
"But perhaps it helped me. I couldn't think and I just had to get on with the job.
"My start wasn't great but it was enough to maintain position, and I think it was key for me to get into a rhythm in the beginning and not lose position [because] then your tyres are dirty and you're always on the back foot.
"That's why I was really happy with our strategy and how we managed that. And actually the pace was really good, the car was great and we made a small adjustment from stint one to stint two.
"I asked a lot of help from the team to use the tools in the car to influence the balance.
"Again, I know that we had a little bit of help from outside with grid penalties and all that. But hey, no one take these points away from us."
De Vries came close to being given a track limits penalty in the race, which he says was down to the "high-speed balance" of his Williams being off and forcing him out wide through the Lesmo section.
"In the beginning of the race I wasn't really happy with the high-speed balance," he added.
"And that kind of forced me always a bit wide. I struggled to turn the car, especially in the two Lesmos, and that was forcing me wide.
"So, with the flap adjust for run two I felt like the balance was in a better window and that helped me to be cleaner and safer on track limits."
De Vries says fellow Dutchman and Italian GP winner Max Verstappen offered him some "kind" advice ahead of his first start, and "appreciates" the support he had from a number of drivers over the weekend.
"He was kind," de Vries said when asked what Verstappen said to him on the grid.
"We had some messaging last night, even this morning. Lewis [Hamilton] congratulated me, they've all been very supportive and very kind.
"Max just came to me and said 'it's going to be fine, it's going to be good, you will be fine. Just have a good start and get through'.
"So, I appreciate the support, and also very thankful to all the fans who voted me as driver of the day."
