Lambert appeared to be on borrowed time as Villa went through a nine-match winless streak in the Premier League, which included six consecutive defeats.
But Hutton's match-minute winner at home to Leicester City on Sunday saw Villa claim back-to-back victories.
Villa are now five games unbeaten and occupy 11th spot in the standings, a run of form Hutton hopes will help ease the fan's frustrations.
"I hope it has eased the pressure on him – we've got a lot of hard games out of the way," said Hutton, who was labelled as the "best right-back" in the Premier League by Lambert following Sunday's triumph.
"I understand the fans' frustrations. They come and want to see you winning and playing good football but we're are five unbeaten now and we want to keep getting better and better.
"If you look at our squad, when we have everyone fit, we rival most teams. Players like Fabian Delph and Ron Vlaar are coming back so we'll have good competition for places.
"I don't see why we can't push up a few more places in the league."
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