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.

It comes as Mr Cahuzac admitted to having had a foreign bank account.
The ex-minister had until now denied a report by investigative website Mediapart that he kept an undeclared Swiss bank account until 2010.
Mr Cahuzac - who gained a reputation as a vocal crusader against the use of overseas tax havens - quit in March.
Mr Cahuzac said he made the admission about the bank account during a meeting with investigating magistrates on Tuesday.
In a statement, he said he had about 600,000 euros (£509,000; $770,000) in the account.
He also asked for forgiveness from French President Francois Hollande and his former government colleagues.
"I was caught in a spiral of lies and lost my way. I am devastated by guilt," the statement said, quoted by the AFP news agency.
"It was an unspeakable mistake to think that I could avoid confronting a past that I wanted to consider behind me. I will now face this reality with all transparency," the statement said.
Mr Cahuzac had earlier insisted that he had done nothing wrong. He was a plastic surgeon before entering politics.
Correspondents say the scandal is embarrassing for the French Socialist government which has made fighting tax evasion a priority, as it tries to cut public debt.
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.
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Former French Budget Minister Jerome Cahuzac has been placed under formal investigation over alleged tax fraud, his lawyer has said.