Morris (2-3) scored two points in the second end and two more in the fourth to edge Koe's (4-1) Calgary rink.
Earlier in the day, Koe cruised to a 7-1 victory over Brandon, Man., native Rob Fowler in Draw 6 action.
Fowler (2-3) was 1-2 on the day, also falling to Winnipeg's Mike McEwen 6-2 in Draw 9. Fowler's only win came against Toronto's John Epping, 8-3 in the afternoon.
Glenn Howard (4-1) of Coldwater, Ont., stayed with the leaders, scoring two points on his last rock in a 7-5 win over Norway's Tommy Ulsrud (1-4) in Draw 7. He also beat Rob Rumfeldt of Guelph, Ont., 5-2 in Draw 6.
Toronto's Mark Kean is also 4-1 after beating Rumfeldt 8-4 and Sweden's Oskar Eriksson (1-4) 8-4.
McEwen (3-2) won both of his Friday games, beating Fowler and adding a 9-6 win over Morris earlier in the day, while Epping (3-2) won his first game 7-4 over Edmonton's Brendan Bottcher (1-4) in another afternoon clash.
Edmonton's Kevin Martin (4-1) downed Winnipeg's Jeff Stoughton 8-2 in five ends in the late draw. Martin lost his first match of the tournament — 7-6 in an extra end to St. John's Brad Gushue — in the afternoon draw.
Stoughton (3-2), also fell 7-5 to Switzerland's Peter de Cruz (3-2) in Draw 7.
Gushue is 3-2 after beating Saskatoon's Steven Laycock 9-4 in Draw 9. Laycock (1-4) downed Winnipeg's William Lyburg (1-4) 7-1 in a morning match.
Rumfeldt (3-2) capped off his three-game day with a 8-1 victory over Sweden's Niklas Edin (2-3).