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Winger Niall McGinn also netted as Northern Ireland recovered from their opening defeat by Poland to remain in contention for a place in the last 16.
"People had written us off, nobody gave us a chance," said defender McAuley. "It's a very special moment.
"We weren't at it against Poland and we wanted to put it right."
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The 36-year-old West Brom player became the second-oldest goalscorer at a European Championship by putting Northern Ireland ahead with a 49th-minute header, before substitute McGinn added the second in injury time.
It was Northern Ireland's first victory at a major tournament finals in 34 years - since beating hosts Spain at the 1982 World Cup - and the result means Michael O'Neill's side can still progress as a best third-placed team at least.
They face Germany in their final Group C game and, with Ukraine having been eliminated, Northern Ireland will qualify for the knockout stages with a win in Paris on Tuesday.
"We let ourselves down against Poland in terms of the intensity we wanted to play at so we had a point to prove to ourselves," said McAuley.
"It was a wonderful team performance and we are delighted for everyone, especially the supporters.
"We have something to play for in our last game against Germany, which is what we wanted. We can take a lot of confidence into that game."
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O'Neill made five changes for his side's second group game, including leaving striker Kyle Lafferty on the bench, and the Northern Ireland boss acknowledge it was a gamble.
"There was some risk attached to changing the team but it was a fantastic performance from the entire side," said the 46-year-old.
"I couldn't have asked any more. The players gave every last ounce of passion and energy and the supporters were magnificent - everything that is good about Northern Ireland was in that stadium.
"We have given ourselves a great chance of finishing third and we will try to nick a point, maybe more, against the Germans.
"I'll let the players enjoy this and then our minds will switch to Germany. We won't underestimate them."

The man's bike was struck by a lorry at 17:15, near a petrol station north of the roundabout junction with the A830.
They warned drivers to expect delays in the area as there are no local diversions.
Officers said at 21:00 on Thursday they expected the road to be closed "for some time" while the emergency services dealt with the aftermath.

While being a fan of "Scandi dramas" like Borgen, Theresa May told the Radio Times she'd tune in to Doctor Who and "a nice Agatha Christie".
Earlier in the day, she'll have a drink with friends before helping churches in her Berkshire constituency who provide lunch for elderly people.
The PM also said she enjoys Strictly Come Dancing, but wouldn't take part.
"I always like to see Doctor Who on Christmas night, if possible, and a nice Agatha Christie to curl up with. David Suchet was a great Poirot - he got him to a T," she said in the Christmas issue of the magazine.
Church has always been an important part of Christmas, she said, as her father was a vicar.
"As a child I had to wait until my father had finished his services before I could open my presents. It felt like a very long wait."
Theresa May: The woman behind No 10
Discussing her TV tastes as a younger woman, the prime minister said she liked watching Diana Rigg and later Joanna Lumley in The Avengers, but never viewed them as role models.
"I have never had a female role model - I've always just got on with doing what I am doing," she said.
Mrs May admitted she had not seen much of former shadow chancellor Ed Balls on Strictly as she tended to be busy on Saturday evenings.
Asked if one day she would be tempted to give the show a try herself, she replied: "I can't dance. It's not a good idea."
Northern Ireland goalscorer Gareth McAuley says Thursday's 2-0 Euro 2016 win over Ukraine - their first at a European Championship - was "massive".
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A cyclist has been killed in a crash on the A82 road to the north of Fort William.
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The prime minister will be curled up on the sofa watching traditional British TV on Christmas Day.