Rock-bottom Villa ended their run of seven straight defeats with a battling 0-0 draw at home to City on Sunday in Garde's first game in charge.
The result closed the gap to safety to five points but Villa survived scares after Fernando hit the bar in injury time and Raheem Sterling and Kevin De Bruyne missed clear chances.
They are still winless since the opening day of the season but former Arsenal midfielder Garde, who replaced Tim Sherwood last week, believes it should boost his new club.
He said: "We have to believe we can be safe
It will be a tough mission even if we have one more point.
"You have to be lucky but you have to work hard, luck is not coming knocking on your door and saying 'look, I'm here'
You create that, the behaviour you need to get lucky
It would have been very disappointing for us not to get a point if they had scored.
"A clean sheet against such a good team, who have such good players all over the field, means we should be a little bit more confident next time
"It was a good occasion to do it
I t was City, and City were full of confidence, they had a brilliant game in Sevilla, so I think it's a good point to work on."
Arsenal's 1-1 draw with Tottenham saw them miss the chance to go two points clear at the top following City's stalemate and the teams are locked on 26 points heading into the international break
The break comes at the right time for City after Wilfried Bony came off in the the first half with a hamstring injury and will be assessed but David Silva, Sergio Aguero, Pablo Zabaleta and Samir Nasri could all return for City when they host Liverpool in two weeks
Boss Manuel Pellegrini said: "I hope they will do their last work with the physio and next Monday will start working with the squad.
"We will see in the way he (Aguero) improves this week but I think he can be (fit after the international break)
It will be easier for us to have a striker.
"Bony has a hamstring injury and we will see on Monday how long he will be
Maybe he must go (with the Ivory Coast) because the doctor of his country must review him but I don't think he will be playing."
City are unbeaten in their last nine games and Pellegrini is pleased with their form despite missing key players.
"It's important for the squad to win the amount of games we did and not to lose in this part of the season when we didn't have important players," he added.
"That's why it's important to still be top of the table, qualifying in the Champions League and still be in the Capital One Cup."
Source : PA