Leitrim led 1-5 to 0-3 at the interval after an Alan Wynne goal on the stroke of half-time.
GMIT narrowed the gap with the second half’s first score, a Johnny Ryan 38th minute free conversion.
After that Leitrim took control and outgunned GMIT with three points from James Glancy, two from Gerry Hickey and another from Paddy McGowan.
Scorers for Leitrim: J Glancy 0-5 (2fs), A Wynne 1-0, R Cox, G Hickey 0-2 each, C Clarke, P McGowan 0-1.
Scorers for GMIT: J Ryan 0-5 (4fs), N Prenty, A Walsh 0-1 each.
LEITRIM: P Farrelly; A Wynne, C Egan, C McGowan; G Reynolds, W McKeon, C Clarke; D Lowe; D Sweeney; D Beck, P McGowan, G Hickey; R Lowe, J Glancy, R Cox.
Subs: C Sheridan for Beck (ht), F McMorrow for McGowan (ht), McGowan for D Lowe (50).
GMIT: P Mannion; M Kelly, A Tierney, D Lee; J McLaughlin, B Sheridan, D Quinn; S Boyle, D McIntyre; E Ward, J Ryan, J Duane; D Cronin, A Walsh, N Prenty.
Subs: P Laffey for Kelly (ht), F McGinley for Boyle (38), S Kavanagh for Cronin (40), N Farragher for Prenty (52), N Lally for Tierney (52).
Referee: O Kelly (Roscommon).