Chuks Aneke struck a 79th-minute winner as the holders, who triumphed over Southend at Wembley in April, struggled against Stanley before eventually making the hat for the second round.
Davis said: "The fans are frustrated because they would expect us to beat Accrington easily, but football isn't that easy - if it was that easy we would win five or six-nil every week.
"We kept the ball well without having any cut and thrust. They worked very hard to stop us, but as the game wore on they got tired and the gaps started to appear.
"We had to be patient which is frustrating for the fans, for us and the players but we deserved to win the game and were glad we got the goal.
"It's a competition that we enjoy playing in and we look forward to the next round."