McGonigle fired home an early opener on three minutes from inside the area.
The former Crusaders and Coleraine striker doubled his tally with a close-range finish on 64 minutes.
Ruaidhri Higgins’ side sit third in the table, one point behind St Patrick’s Athletic but with a game in hand.
Sligo Rovers are level on points with Derry in second place but are two ahead of goal difference thanks to a 3-1 victory over Finn Harps.
After early chances for Drogheda for Dayle Rooney and Sean Roughan, both saved by Brian Maher, McGonigle opened the scoring on three minutes with a powerful finish from inside the area.
Joe Thomson and Daniel Lafferty both denied by Sam Long in the Drogheda goal, who lost key midfielder Darragh Markey just before half-time.
Darragh Nugent fired wide of the Derry goal in a rare way attack, and McGonigle missed a chance to double both his and Derry’s tally when he rolled the ball wide from Shane McEleney’s pass.
The second half was a sluggish affair and McGonigle’s second goal was a rare highlight. The forward netted his fourth of the season when he slotted home from close range on 64 minutes to double Derry’s advantage after a scrappy passage of play.
The returning Michael Duffy was introduced by Higgins for his second debut at the Brandywell but the winger lasted only 10 minutes when he hobbled off after a tackle by Keith Cowan in a blow for the Candystripes.
Duffy’s injury took any remaining flow out of the game but Derry had enough in hand to secure their third win in five matches.
Next up for Derry is a home match against St Patrick’s Athletic this Friday, March 18 (kick off 7.45 pm).
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