York's Patrick McLaughlin carved out City's best opportunity in the 11th minute, sending a 20-yard shot on the run just wide after splitting the home defence.
A great chance, in the 17th minute, fell to young forward Paris Cowan-Hall, who clipped the ball over Ingham but then swiped at the bouncing ball to send his shot wide of an unguarded goal.
Argyle took the lead in the 44th minute following Luke Young's corner from the left.
Goalkeeper Michael Ingham failed to claim the ball and in the ensuing goalmouth scramble the ball ricocheted off City left-back Jamal Fyfield and in, helped over the line by Plymouth target man Rhys Griffiths.
Substitute striker Nick Chadwick claimed the second goal in the 89th minute after Ingham made a superb save to keep out Young's thumping shot from 18 yards.
Argyle's last league win was 3-1 over Rochdale on October 20.