Holders Dundalk made eight changes after Thursday's superb Europa Cup win over Maccabi Tel-Aviv but they led 2-0 at the break with clinical finishing.
A header from Dean Shiels was followed by Ciaran Kilduff's composed volley.
Rory Patterson poked in before substitute Curtis rifled home to set up a Brandywell replay on Tuesday.
Dundalk boss Stephen Kenny rung the changes after Dundalk beat Maccabi 1-0 to become to the first Irish side to secure a group stage victory in European football.
Derry hoped to take advantage of any tiredness and the visitors made an impressive start with Patterson's glancing header bringing out out an excellent save from Gabriel Sava.
City were on top but Dundalk netted with their first meaningful attack on 21 minutes - Alan Keane jinked through on the right and his cross was met by Shiels with a diving header.
Shiels, the son of Derry boss Kenny, suffered an elbow injury as he scored and he was forced off with former Candystripes midfielder Patrick McEleney his replacement.
McEleney created the second goal in first-half added time, a dinked pass finding Kilduff and striker flicked the ball beyond keeper Gerard Doherty.
The two players combined again five minutes after the restart with Kilduff, who netted the goal against Macabbi, seeing his header smash against the woodwork from a McEleney cross.
Derry kept battling and they were rewarded when Patterson pounced to tap in after Sava failed to hold Aaron McEnuff's deflected shot on 65 minutes.
Curtis netted a deserved leveller, a fine first touch setting him up to blast low inside the post from 20 yards.
Dundalk had a chance to to win it in injury-time winner but the unmarked David McMillan sent his effort high and wide.
Ronan Curtis scored an 86th minute equaliser as Derry fought back from two goals down to draw with Dundalk in the FAI Cup semi-final at Oriel Park.