The Candystripes had gone into the game bottom of the Premier Division table after losing their opening two matches.
Striker Curtis struck a fine half-volley in the 60th minute when the ball fell to him following a corner.
Bohs had a great chance to equalise when Conor McDermott fouled Kevin Devaney, but Eoghan Stokes’ penalty was saved by Derry keeper Ger Doherty.
After three away games, Derry will play the first fixture at their redeveloped Brandywell stadium on Friday night when they host Dundalk who are managed by former Candystripes boss Stephen Kenny.
The ground, where Derry have played since 1928, now boasts a 1,100-seat stand, named in honour of the club’s former striker Mark Farren who died in 2016 after suffering a brain tumour.
While Derry will go into that match boosted by their win at Dalymount Park, Dundalk’s confidence will also be high as they hammered Limerick 8-0 on Tuesday night.
Bohemians had most of the possession in the first half against Derry, and keeper Doherty did well to when Stokes was through on goal.
City had their chances too, with Curtis shooting straight at keeper Shane Supple.
After the interval, former Northern Ireland striker Rory Patterson dragged a shot wide for Derry.
But then came the goal by Curtis which edged the visitors in front.
In the 85th minute it looked like Derry might be denied the win as McDermott’s pull back on Devaney resulted in referee Rob Harvey pointing to the spot.
But City keeper Doherty dived low to his right to keep out the spot-kick from Stokes.
Next up for Derry is its first home game of the season this Friday night at the revamp Brandywell stadium against Dunbalk.