Villa are seven points away from safety in the Premier League and face the prospect of being bottom at the turn of the year, as Leicester were in 2014-15.
The Foxes were three points off 19th-placed Burnley going into January 2015 but picked up 28 points from their next 19 games to finish 14th.
They replaced manager Nigel Pearson with Claudio Ranieri in the close-season and have enjoyed an incredible start to the campaign, leading the table by two points after 15 matches.
Ahead of Garde's meeting with former club Arsenal on Sunday, he told reporters: "Leicester were in the same situation [as Villa]. Why can't we do that?
"I was pleased with the spirit of my players [in the 1-1 draw] at Southampton. I was very happy to see the fighting spirit.
"I saw the Bournemouth and Newcastle results but you concentrate on what you can control," he added, referring to surprise wins for two fellow strugglers.
Garde has recalled Jack Grealish to the first-team squad for Sunday, the midfielder having been dropped after visiting a nightclub shortly after Villa's 4-0 defeat to Everton last month, although defender Micah Richards is doubtful.
"Micah Richards is currently struggling with a knee problem - a slight one - but he is still struggling," Garde said.