Hanover hit the half-time interval on level terms but eventually came out on top against Fivemiletown United.
Bob Radcliffe Cup progress arrived thanks to goals by Justin Bradley, Scott McCullough, Jamie Rooney and Chris Elliott.
Hanover finished with a 5-0 success despite entering the break with a blank tally.
Bradley kicked off the scoring and McCullough added to his recent run of strong form.
Rooney - a summer signing from Tandragee Rovers - increased the advantage.
Elliott converted two penalty kicks as Hanover marched on at the expense of Fivemiletown.