In the second-half Gabriel Jesus, Mahrez, Nicolas Otamendi and substitute Leroy Sane completed the rout, with Ilkay Gundogan registering four assists. Barnet is now managed by Darren Currie, the nephew of Sheffield United great Tony Currie.
Watford booked their spot in the fourth round after crusing to a 2-0 win over Woking.
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.
It had seemed that visitors Leicester, seventh in the Premier League and top-flight champions in 2016, would leave Rodney Parade with a draw when Rachid Ghezzal's shot eight minutes from time cancelled out Jamille Matt's 10th-minute header.
Lakers vs. Thunder Recap: The good, the bad and the Ball
Justin Holiday scored three points and only made one of his seven shots from the field as the Bulls lost to the Magic 112-84. Although Chris Paul is out, the Rockets may be able to will themselves to another victory against the reigning champions.
Sam Surridge and Callum Lang got the goals to send Claudio Ranieri's charges packing.
County were back in front within three minutes, however, as Marc Albrighton handled and Amond coolly converted from the spot.
He said: 'I'm probably the proudest man in Oldham.
City scored early through Raheem Sterling but had to wait until shortly before halftime to add to their total with Phil Foden scoring his first home goal for the club and Semi Ajayi turning Kyle Walker's cross into his own net.
Championship leaders Leeds United suffered a 2-1 defeat at Queens Park Rangers, joining second-placed Norwich City who were beaten by Portsmouth on Saturday.
Sheffield United began the day 84 places above Barnet but were outplayed by their lowly opponents for whom former Tottenham trainee Shaquile Coulthirst won it with a 21st-minute penalty after Richard Stearman brought down Ephron Mason-Clark.