The only goal arrived in the 58th minute when Palace goalkeeper Wayne Hennessey allowed a header from former England centre half Joleon Lescott to slip through his hands and into the net.
"That was very important for the team, for the club, for everybody," Garde told Sky Sports television. "We have suffered so much over previous weeks but today we are very happy.
"I think we deserved to win because we put them under pressure and that's what we wanted. We created many good opportunities and always tried to attack -- it's a good win for us."
Villa are still eight points from safety despite having registered their first league victory since the opening day of the season.
The home team seemed set for another 90 minutes of struggle when Palace's Wilfried Zaha struck the woodwork inside the opening minute but Garde's men held firm.
"We were helped after 20 seconds by the post, I don't forget that but that's the game," said the Frenchman, who took over in early November following the sacking of Tim Sherwood.
"I think we made our luck today by being very aggressive on the ball and wanting to attack. I'm very pleased for the players because they made huge efforts today ... now we have to build on this win."
Garde's counterpart Alan Pardew described seventh-placed Palace's performance as one of their poorest of the season.
"A lot of things went wrong today," said Pardew. "We lacked energy and we lacked quality.
"Villa deserved the victory because they worked so, so hard. They didn't really give us a moment's rest, they really had a go.
"We couldn't find any cohesion and it was a disappointment, no doubt about that," added Pardew. "We had no cohesion in the team and that's something we're renowned for."
(Writing by Tony Jimenez; Editing by Nick Mulvenney)
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