Obuya slammed 57 off 89 balls, including five fours and a six, while Odoyo struck four fours to lend respectability to the Kenyan total after some disciplined Zimbabwe bowling.
Zimbabwe, still rankled by a seven-wicket defeat in the World Cup Super Six despite an impressive overall 12-1 win-loss record against the Kenyans, made an early an breakthrough when Heath Streak snared Kennedy Obuya for a duck.
But David Obuya featured in two useful partnerships to steady the innings. He added 52 from 78 balls with young Brijal Patel (18) and 59 off 72 balls with skipper Steve Tikolo before falling to Raymond Price in the 28th over.
Hard-hitting Tikolo contributed another useful 37 before hooking the ball straight into the hands of Douglas Marillier off Price, who replaced Travis Friend from the side that faced Pakistan on Thursday.
Hitesh Modi made an unbeaten 27 off 36 balls in the company of Peter Ongondo (10 not out) to ensure a decent score for their side.
Streak returned completed impressive figures of one for 28 off 10 overs, including three maidens, while Price took two for 36.
Kenya, the first non-Test playing nation to qualify for the semifinals of the World Cup, are playing their first match in the four-nation event which also features Sri Lanka and Pakistan.
Zimbabwe, beaten by Pakistan in their opening match, face a modest target of 4.5 runs an over under lights to record their first points in the tournament.
Scoreboard
KENYA:
K.O. Obuya c Taibu b Streak 0
D.O. Obuya c Rennie b Price 57
B.J. Patel b Hondo 18
S.O. Tikolo c Marillier b Price 37
T.M. Odoyo c Marillier b Hondo 46
M.O. Odumbe c and b Rennie 17
H.S. Modi not out 27
P.J. Ongondo not out 10
EXTRAS (LB-1, W-12) 13
TOTAL (for six wkts, 50 overs) 225
FALL OF WKTS: 1-3, 2-55, 3-114, 4-131, 5-161, 6-201.
DID NOT BAT: C.O. Obuya, M.A. Suji, A.O. Suji.
BOWLING: Streak 10-3-28-1; Blignaut 4-0-27-0 (8w); Hondo 8-0-37-2; Price 10-0-36-2; Flower 3-0-20-0; Marillier 6-0-30-0 (3w); Ervine 4-0-26-0; Rennie 5-0-20-1 (1w).
ZOMBABWE: C.B. Wishart, D.A. Marillier, G.W. Flower, D.D. Ebrahim, G.J. Rennie, T. Taibu, A.M. Blignaut, H.H. Streak, S.M. Ervine, R.W. Price, D.T. Hondo.
UMPIRES: B. Doctrove (West Indies) and A.V. Jayaprakash (India).
TV UMPIRE: K.C. Barbour (Zimbabwe).
MATCH REFEREE: A.M. Ebrahim (Zimbabwe).—Reuters/AFP