Stephen Carson put the Glens ahead but goals on either side of half time from George McMullan put Cliftonville 2-1 up.
Then late goals from Liam Boyce and Tomas Cosgrove ensured a confident win for Tommy Breslin's side.
Cliftonville 4-1 Glentoran
Cliftonville: McMullan 43 & 49, Boyce 80, Cosgrove 90+3
Glentoran: Carson 13