Kenya, chasing 325 for an unlikely win, were held to 264 for six for their fifth defeat in as many group matches.
Kenya faced a daunting chase considering their previous best total in this group was the 198 they'd made last time out in a five-wicket defeat by Canada.
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Opener Alex Obanda struck fast bowler Shaun Tait for two superb sixes, flaying him over point and driving him high over long-on to the delight of neutral fans in a crowd of 13,600.
But when he was bowled for 14, swinging across a Tait full toss, Kenya were suddenly 46 for three when a mix-up between the Obuya brothers saw David run out.
Collins Obuya (98 not out) and Tanmay Mishra (72) kept Australia at bay with career-best innings in a fourth-wicket stand of 115, although the pair's run-rate never suggested an upset win.
Mishra made a 63-ball fifty featuring a six and six fours. It was his second half-century in as many innings following his 51 against Canada.
The pair played Australia's lead spin duo of Steven Smith and Jason Krejza with relative ease, striking several boundaries off the slow bowlers whose combined 14 wicketless overs cost 72 runs. Mishra fell for a career-best 72 after a needless run out saw him beaten by Michael Clarke's direct hit from backward point.
Collins Obuya completed his fifty in style with a huge six over long-off against Shane Watson and later pulled him high above square leg.
He needed three off the last ball, from Tait, to get a hundred but could only manage a single. In all he faced 129 balls with three sixes and nine fours.
Earlier Australia, now unbeaten in 33 matches at the World Cup dating back to 1999, lost three wickets for 16 runs to slump to 143 for four. But a stand of 114 between vice-captain Clarke (93) and the returning Michael Hussey (54) helped Australia finish on 324 for six.
Clarke, on the ground where he made a hundred on Test debut against India in 2004, struck a six and seven fours during a composed 80-ball knock before he holed out off fast bowler Nehemiah Odhiambo, who took three for 57.
Opening batsman Brad Haddin, dropped twice off tough chances early in his innings, made 65 and helped skipper Ricky Ponting add 89 for the second wicket.
Haddin was caught at deep midwicket by Rakep Patel off Kenya captain Jimmy Kamande, who took two for nine in eight balls with his off-spinners. Ponting was lbw to leg-spinner Collins Obuya for 36.
Hussey, in at the fall of the fourth wicket, was playing his first match of the tournament after recovering from a hamstring problem and getting a late summons as a replacement for injured fast bowler Doug Bollinger.
The 35-year-old left-hander, in for dropped younger brother David, demonstrated customary calm at the crease during a 43-ball innings.
S Watson c Ouma b Odhiambo 21
B Haddin c Patel b Kamande 65
R Ponting lbw b C Obuya 36
M Clarke c Patel b Odhiambo 93
C White b Kamande 2
M Hussey c D Obuya b Odhiambo 54
S Smith not out 17
M Johnson not out 12
Extras (b2, lb5, w16, nb1) 24
Total (6 wkts, 50 overs) 324
Did not bat: B Lee, J Krejza, S Tait
Fall of wickets: 1-38 (Watson), 2-127 (Haddin), 3-131 (Ponting), 4-143 (White), 5-257 (Hussey), 6-304 (Clarke)
Bowling: Odoyo 10-0-50-0 (1w); Otieno 8-0-75-0; Odhiambo 10-1-57-3 (2w); Ngoche 8-0-56-0 (1w); Kamande 10-0-46-2 (2w); C Obuya 4-0-33-1 (1nb, 1w)
M Ouma c Haddin b Lee 4
A Obanda b Tait 14
C Obuya not out 98
D Obuya run out 12
T Mishra run out 72
T Odoyo b Tait 35
R Patel run out Tait 6
J Kamande not out 0
Extras (b2, lb6, w12, nb3) 23
Total (6 wkts, 50 overs) 264
Did not bat: N Odhiambo, E Otieno, J Ngoche
Fall of wickets: 1-12 (Ouma), 2-21 (Obanda), 3-46 (D Obuya), 4-161 (Mishra), 5-247 (Odoyo), 6-263 (Patel)
Bowling: Lee 8-1-26-1; Tait 8-0-49-2 (2nb, 5w); Johnson 8-1-40-0 (2w); Smith 6-0-36-0; Krejza 8-0-36-0; Clarke 5-0-21-0; Watson 7-0-48-0 (1w).
Result: Australia won by 60 runs