Nicky Harrison sent the Staffordshire crowd wild with a 23rd-minute goal but veteran David Eyres spared the Latics' blushes.
Signs of an upset were evident from the kick-off with Neil Aulton twice going close to giving the non-league side the lead.
Chasetown finally went ahead when Harrison unleashed a 25-yard strike which ricocheted off keeper Chris Day and trickled over the line. However, the lead lasted less than 10 minutes as 41-year-old Eyres equalised following a neat one-two with Andy Liddle.
The non-leaguers should have restored their lead just after the break. However, an alert Day thwarted Damien Whitcombe's curling free-kick.
Chris Porter and Eyres both spurned chances for Oldham to seal the win as the League side's superior fitness edged the closing exchanges.
Chasetown could have another home tie, against Brentford, in the next round if they can win the replay at Boundary Park.
Non-League Burscough could face a trip to Peterborough as reward for knocking League One Gillingham out of the Cup.
The UniBond League Premier Division side beat the Gills 3-2 on Saturday and will now travel to League Two Peterborough or Conference strugglers Burton Albion in round two. Tamworth, who overcame Bournemouth, face a trip to League One Hartlepool.
One non-League team who may fancy their chances of reaching the third round - and earning a possible date with a Premiership club - are Conference leaders Grays Athletic, who will face League Two strugglers Mansfield.
Chippenham and Worcester, who drew 1-1, will replay for the right to host League One high-flyers Huddersfield.
Harrogate Town will host Notts County if they win their replay against Torquay while Weymouth could could play Chester if they beat Nottingham Forest. Cheltenham will host Eastbourne Borough or Oxford.
Two ties ensure there will be some non-League presence in the third round with Southport or Woking playing Northwich and Nuneaton facing Histon.