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Houghton told BBC Sport: "The players have a healthy respect for him. And if the fans see progress and a bit of belief and spirit, he'll be fine."
Cascarino added: "There's huge respect for Steve in Ireland and he knows how to get the best out of players."
Staunton, who won 102 international caps, is currently Paul Merson's assistant at League One side Walsall.
He played in three World Cup finals, with his final game being the second-round penalty shoot-out elimination at the hands of Spain in South Korea in 2002.
However, Houghton and Cascarino, who played alongside Staunton for the Republic, say the Football Association of Ireland should appoint former England boss Sir Bobby Robson as his assistant, as has been mooted.
"Sir Bobby has been it, seen it and done it," said Houghton. "Steve can lean heavily on him, especially when it comes to the media.
"But once he gets to grips with it, Steve will be his own man and want to pick the team.
"A lot of people will level the fact he's inexperienced against him, but other nations have done the same thing.
"Marco van Basten got Holland qualified for the World Cup this year and Jurgen Klinsmann was plucked from nowhere and is manager of the host nation [Germany]."
And Cascarino agreed that Staunton's inexperience could be overcome and said that one of his main tasks would be restoring team spirit.
"It's very tough to walk out of your playing days and straight into management," said Cascarino. "But having someone alongside him will help him enormously.
"He'll want to get that typical fighting spirit back and get a great atmosphere going. There was a lot of paranoia towards the end of Brian Kerr's reign and that spilled over onto the pitch.
"The Irish public will be a bit dubious because of his inexperience, but they'll be pleased because it's an Irishman who'll be leading their country."
However, both Houghton and Cascarino sounded a note of caution.
"The only way you can change opinions is by getting results," said Cascarino. "He'll get at least one qualifying campaign - but he's certainly got to qualify for Euro 2008 or the 2010 World Cup."
Houghton added: "If he's bringing in younger players, they may not be ready for Euro 2008. As a manager, you're judged on results and if we're coming bottom of the group, there are going to be problems."