Tabloid newspapers the Irish Daily Mirror ran with the headline "We're back", adding that the win meant Staunton's job was safe for the time being and kept Ireland's chances of qualifying alive.
Ireland have 13 points in Group D, equal with leaders Germany and the Czech Republic, but have played more games.
Staunton, who took over as Ireland manager in January 2006, has weathered severe media criticism and repeated calls for him to quit after a humiliating 5-2 defeat by Cyprus last October and February's last-gasp 2-1 victory over San Marino.
Even last weekend's 1-0 victory over Wales, when Ireland allowed their opponents to control a goalless second half, did little to pacify many critics.
Vincent Hogan, Irish Independent columnist, wrote that in their "lust for victims", some in the media appeared to have forgotten that, even before Wednesday's victory, Ireland had chalked up three wins in a row.
"Seldom can successive wins in competitive internationals have decanted a more venomous storm of criticism," Hogan said.
"The worst of it was jack-booted and tasteless."
Writing in the Irish Examiner, Liam Horan, television analyst, observed that commentators had shown "an assassin's sense of charity" prior to the match and that even afterwards some had welcomed the win only "grudgingly".
The Irish Sun said former England manager Bobby Robson, who was hired as a mentor to the relatively inexperienced Staunton, had called on critics to "Lay off Stevie!".
"Hopefully, now we will get a period of calmness, some sensibility, something constructive, intelligent, fair, something reasonable for the next six months," the paper quoted Robson as saying.