Winger James Henry tormented the Cobblers - who nervously sit three points above the League One drop zone - all evening and his quality made the difference after 35 minutes.
The on-loan Reading star beat Danny Jackman down the right and his superb low cross left Alexander with the simple task of tucking home his ninth of the season from three yards.
Henry then nearly added the goal his performance deserved on the hour, somehow heading David Martin's perfect cross inches wide when it was easier to score.
The Lions had dominated the opening half without creating many chances until Jimmy Abdou failed to connect with Marc Laird's measured cross on the half hour.
The visitors, clearly missing the power of suspended targetman Adebayo Akinfenwa, offered little in the attacking third, with Abdul Osman's half-volley after 41 minutes their only real opportunity.