Gauci played a key role in securing a point in Tuesday’s goalless draw against Barnsley, saving a second-half penalty as Vale leapfrogged Northampton Town into 23rd place.
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It was the first time Vale had not finished bottom since losing to Plymouth on 22 November.
But their position remains dangerous, with a 15-point gap to safety, with a maximum of 18 points available.
However, Gauci told BBC Radio Stoke that not being at the bottom gives them something to hold on to.
“I think it gives us a lot of motivation,” he said.
“Everyone has been talking about the schedule for this end of the season and these last three weeks are probably the busiest period of the year.
“I am very motivated and I want to do well. We are all competitive. We all want to win football games.
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“One day at a time and we continue to focus on the team above us and work to catch them.
“It’s a great motivation and it’s going to push us forward these last two and a half to three weeks.”
Vale’s success in the FA Cup has contributed to fixture congestion which has resulted in them playing eight times in April before the final game of the season on 2 May.
They play the second of four games in nine days at Peterborough on Thursday, where a defeat would leave them on the brink of an immediate return to League Two.
Gauci, on loan from Aston Villa, has made 28 appearances for Vale after first replacing Marco Marosi five games into the season.
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Despite his struggles, the 25-year-old Australian said competing for regular action has been good for him.
“That’s why I wanted to come here to challenge and get minutes and help the team and the club,” Gauci said.
“It’s the same for every player. People are quick to tell me, ‘You’ve got a lot of time on your side,’ but I don’t really see myself that young. I want to play well and consistently.”