Right-back Douglas capped a fine all-round display when powering home Ryan McEvoy’s free kick with a bullet header on 88 minutes.
It was a win Bray fully deserved as they dominated for good spells and would have won by more but for Micheál Schlingermann.
The Drogheda keeper tipped over a 30-yard drive from Graham Kelly before making a sublime save when somehow getting a touch to deflect Kelly’s close-range shot onto the bar.
When Schlingermann was cleanly beaten, from McEvoy’s stinging 25-yard effort in first half stoppage time, his right-hand post came to the rescue.
“It’s something we’ve been working on in training,” said boss Cooke of Douglas’ winner.
“He’s a big, strong boy. He should probably be getting more goals from set pieces.
“Hopefully, that will give him a bit of belief, that he can get on the end of things. It was nice for him to score his first goal.”
Bray now face a St. Pat’s side looking for a record ninth league win on the spin.
“I was in Tallaght on Friday night and I thought it was an absolutely fantastic performance by St. Pats,” said Cooke of St. Pat’s 2-0 win at Shamrock Rovers.
“It’s going to be a big ask for us. But when you’ve only one defeat in six games, our confidence will be high as well.
“With the form we have and the form they’re in, we’ll look forward to a good game.”
Cherrie; Douglas, McNally, Cooney, Barker; Cassidy, Kelly, Sullivan, McEvoy, McGlynn (Onwubiko, 57); Lyons (Scully, 79).
Schlingermann; Daly, Buckley, O’Reilly, Marks; Whelan (Pender, 69), Maher, Thornton (McKenna, 90+2), Mulenna; Brennan (Bayly, 74); Duffy.