Harrison boosted Chasetown's giant-killing hopes with a 23rd-minute goal which put the Midland Alliance club into a well deserved lead against a club which once graced the Premiership.
But their hopes of progressing were dented by the oldest player in the Football League. David Eyres, who will be 42 in February, rescued Oldham with an excellent 31st-minute equaliser.
With no less than 133 clubs separating the two clubs the result was an outstanding achievement for Chasetown.
Manager Charlie Blakemore said afterwards: "I am delighted for my team. I am very pleased that they put on a decent show in front of the television cameras.
"We were always in the game and the players were a credit to themselves."
Oldham boss Ronnie Moore, who admitted that previously he had no idea where Chasetown was let alone what competition they played in, said: "We were fortunate that we did not go out. But this is the FA Cup.
"We will be better prepared for the replay as we now know how Chasetown operate. We will also be playing on a much better pitch."
Chasetown, to their credit, never allowed Oldham to settle and the Football League outfit could not raise the tempo to destroy the little non-league club.
Harrison gave the game a lift with a goal which he scored with a low long-range effort which went in via the post and the goalkeeper's foot.
Encouraged by this the Scholars always battled hard. But veteran Eyres came to Oldham's rescue when he cut in from the left and notched the equaliser with a low left-foot shot.
The closest Oldham came to winning occurred just before the break when Chris Slater raced back to head clear from just under the crossbar to keep out a Gareth Owen header.
A determined Mark Tierney attempted to inject some life into Oldham but Chasetown held on to enjoy another taste of Cup glory.