However, Corinthian should have taken the lead shortly after the half-hour. Malik Thomas capitalised on a defensive mix-up, and after his first shot cannoned back off the post, his rebound was brilliantly cleared on the line by centre-half Jack Tucker.
After Chapman, Stevenson, Lawford and Ryan Huckle had all come close to an opener either side of the break, and Gills goalkeeper Will Sykes had produced a good save to deny Conor Johnson, the Gills went ahead. Lawford's perfectly-placed cutback found Stevenson, who fired home from six yards.
Goal scorer turned provider for the Gills' second on 67 minutes, as Stevenson's cross from the right found Mbo, who headed home at the back post. Lawford completed the scoring 20 minutes from time, firing home confidently on the turn.
Lawford blazed over following a quick break from a Corinthian corner, and Stevenson also shot narrowly off-target as the Gills searched for a fourth. However, the final goal of the game came on 92 minutes, as Johnson weaved through the home defence and fired past Sykes.
Nonetheless, the Gills progress to round two, for which ties have to be played by Saturday, November 19. Full information will follow on the GFC website in due course.
Gillingham: Sykes, Divine (Conway 59), Huckle, O'Mara (C), Tucker, Simpson, Oldaker, Stevenson, Chapman, Lawford (Woods 78), Mbo (Hlabi 78)
Unused subs: Catherall [GK], White
Corinthian: Johnson, Holland, Mills, Smith, Pearce, Styles, Thomas, Butcher, Johnson, Housego, Clarke
Subs: Earl [GK], Bradford, Smith, Oladipupo, Adekoya
Referee: Simon Finningan