The Queenstown skier followed her eighth placing in super G with another eighth, out of 76 athletes, in her specialist giant slalom event.
She finished 0.8sec behind winner Federice Brignone and just 0.18sec off the silver medal position in an exceptionally tight competition.
"There have been so many emotions for me in the past month," Robinson said.
"It was just quite a big day. I really wasn’t sure what to expect today.
"I think GS has been a bit of a question mark for me the past two months — from having such a good start to the season and then I was really struggling a lot, especially in the last race. So, today I really didn’t know what was going to happen.
"I felt a lot better on my skis but I didn’t really feel like I was going to be in the running for a medal. Then to have it be so close and let it slip away ... it was quite sad but I’m proud at the same time."
Conditions at the Tofane Alpine Skiing Centre in Cortina d’Ampezzo were ideal, with clear blue skies overhead and cold temperatures delivering the hard and fast snow conditions needed for ski racing.
Only racers who completed their first run got to take a second run, so there was no margin for error.
Starting from bib six, Robinson took a few turns to settle into her rhythm.
A couple of small mid-course errors left her a fraction late into the gates, but she recovered well and showed strong control over the finish jump.
At the end of the first run, Robinson was sitting in 10th place with a time of 1min 4.32sec, 1.09sec behind Brignone.
She powered out of the gate for her second run and looked strong and confident on course, charging her way through and searching for speed at every turn, but could not quite beat the clock, posting 1min 9.38sec.
"I was pretty disappointed," Robinson said, describing the moment when she first crossed the line to see that she had missed out on a medal spot.
"I came down, and I saw I was in fourth (at that point) and only 12-hundredths behind. It was hard, but then watching the rest of the race play out, it actually ended up a bit better than I thought."
Robinson was competing at her third Olympics after a mixed world cup season that started with spectacular results before her form dipped and she had a major crash.
"It’s been a lot of work for everyone," coach Tim Cafe said.
"Last Olympics were really challenging for Alice and I’m so happy that she’s back to skiing well again.
"What we’ve been through in the last month ... it’s going to take a bit of time to reflect on that."
Italian star Brignone again delighted her home crowd as she backed up her win in the super G with gold in giant slalom.
Sara Hector (Sweden) and Thea Louise Stjernesund (Norway) posted exactly the same times on both their first and second runs, meaning they shared the silver medal position.
American great Mikaela Shiffrin finished 11th.
— Allied Media
