Wrexham's supporters, making a round trip of nearly 600 miles, had little to cheer as their side rarely threatened.
The Whites dominated after ex-Newport striker Bird turned-in from close range in the 18th minute after Kane Richards' set-up.
Chris Kinnear's side are sixth in the National League after three games.
Wrexham manager Dean Keates told BBC Radio Wales: "We came with a game plan and for the first 15 minutes it was working very well and ideally we should have been 1-0 up in the opening minutes.
"After they scored it wasn't the reaction we were after if I'm honest. Second half it wasn't anywhere near what it should have been."
Second Half ends, Dover Athletic 1, Wrexham 0.
Substitution, Dover Athletic. Kadell Daniel replaces Kane Richards.
Shaun Pearson (Wrexham) is shown the yellow card.
Substitution, Dover Athletic. Tobi Sho-Silva replaces Ryan Bird.
Substitution, Wrexham. James Hurst replaces Marcus Kelly.
James Jennings (Wrexham) is shown the yellow card for a bad foul.
Mitch Brundle (Dover Athletic) is shown the yellow card.
Substitution, Wrexham. Ntumba Massanka replaces Scott Boden.
Substitution, Wrexham. Paul Rutherford replaces Mark Carrington.
Kane Richards (Dover Athletic) is shown the yellow card for a bad foul.
Second Half begins Dover Athletic 1, Wrexham 0.
First Half ends, Dover Athletic 1, Wrexham 0.
Femi Ilesanmi (Dover Athletic) is shown the yellow card for a bad foul.
Goal!  Dover Athletic 1, Wrexham 0. Ryan Bird (Dover Athletic).
First Half begins.
Lineups are announced and players are warming up.
Ryan Bird's first-half goal gave Dover Athletic a second win in five days as they beat Wrexham at the Crabble Athletic Ground.