Boyle struck the only goal in the 18th minute after lobbing over Graham Doyle following a moment of hesitancy in the Wexford defence.
Derry needed fine saves from Gerard Doherty to stay ahead as the keeper denied Conor O'Keeffe and Paul Murphy.
Josh Daniels also headed a Gary Delaney effort off the line in the second half as Derry endured some nervous moments.
Derry did go close to adding to their lead in the final five minutes of the game as Aaron McEneff clipped the Wexford crossbar with a long-range shot and substitute Jordan Allan fired just wide.
Kenny Shiels' side suffered a blow in the second half as goalscorer Boyle was forced off by an apparent hamstring injury.
With leaders Dundalk beating St Patrick's Athletics 2-0, the Candystripes remain nine points off the pace but they have edged to within six points of second-placed Cork City who were held to a 0-0 draw by Shamrock Rovers in Dublin.
Nathan Boyle's first-half goal proved enough to give Derry City victory over Wexford Youths at the Brandywell.