The former Bohemians and Longford Town midfielder has agreed to extend his stay at the club for another season, while the 22-year-old goalkeeper has also signed a new deal
The former Bohemians and Kildare County midfielder joined the Drogs from Longford Town in 2012 and was part the squad that finished runners-up in the Airtricity Premier Division that season.
Boynesiders boss Robbie Horgan said that he was "extremely pleased" to have tied down Brady ahead of the forthcoming campaign and suggested that the 28-year-old would be an important player for the club.
"I'm extremely pleased to have secured this signature. Cathal is very talented midfielder and a very positive influence both on and off the pitch," Horgan said.
"Once again I have to point to the fact that having the right character and mentality is hugely important in the squad that I am building and Cathal is one of the players that I had hoped would remain with us next season.
"I know that he will give 100 per cent and I believe that his perseverance and commitment will be a contributory factor in how we measure up next season."
The Drogs have also confirmed the re-signing of 22-year-old goalkeeper Micheal Schlingermann.
"I had a fantastic season in Drogheda in 2013 and it feels right to continue my career here," said Schlingermann after signing. "I am really happy with the professional set-up that Robbie is putting in place.
"[Drogheda United goalkeeper coach] Dermot O'Neill is known throughout the league as one of the top goalkeeping coaches and I feel privileged that I will be mentored by him next season."