Managerless Oldham shocked Premier League strugglers Fulham with a 2-1 win at Craven Cottage while Sheffield United were dumped out of the FA Cup by non-league Barnet on Sunday.
Jamille Matt headed home for the fourth tier side in the 10th minute, putting them on track for a huge upset, only for Rachid Ghezzal to break the home side's hearts with an equaliser in the 82nd minute.
Newport's 2-1 victory against Leicester was their first FA Cup win against a side from the top tier of English football since beating Sheffield Wednesday in the third round of the 1963-64 edition, ending a run of seven matches against such opponents without a win (D3 L4).
The win follows on successes in last year's competition when Newport beat Walsall and Leeds United and came within eight minutes of dumping Tottenham Hotspurs out of the competition.
Sheffield United have lost a home tie in The FA Cup to non-League opposition for the first time since Darlington beat them 1-0 in January 1911.
Bidwell then headed in the victor with 15 minutes to go from a set-piece.
"It is one you have to savour", Oldham boss Pete Wild said. I can't think of many others.
"It's Roy of the Rovers stuff".
"It feels fantastic, a proud moment for me, the boys and the club", former Barnet player Currie, who took over from John Still last month, told the BBC.
Victory was particularly sweet for Currie whose uncle Tony was an England worldwide and was so revered as a player at Sheffield United that a stand at the club's Bramall Lane home is named in his honour.
"I wouldn't say they deserved it, but they battled to the end".
It turned into a rout after the interval, Gabriel Jesus netting from Sterling's precise cross, with Ilkay Gundogan then turning provider for City's final three goals - Riyad Mahrez, Nicolas Otamendi and Leroy Sane getting in on the act.
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