The pair put on 137 for the fifth wicket after the West Indies, electing to take first strike on the dry wicket, were reduced to 63-4 by the 15th over.
Lara, who dropped himself to number six, made up for a hesitant start to flay two sixes and seven boundaries off the world champions in his 71 before hurting his back while playing a shot.
He called for treatment soon after smashing fast bowler Brett Lee for a six and managed to add only four more runs before spooning a catch to the covers off Glenn McGrath.
Morton remained unbeaten on 90, a knock studded with seven fours and a sweetly-timed six over long-on against McGrath in the penultimate over.
It is the first match for both teams in the main tournament. England and hosts India are the other two teams in group A where the top two will qualify for the semi-finals.
The centre wicket at the Brabourne stadium played better than the dusty side tracks provided for the previous two matches that were described as "sub-standard" by South African captain Graeme Smith.
The ball still did not come on to the bat to facilitate strokeplay, forcing the batsmen to grind for their runs.
The West Indies, bowled out for 80 by Sri Lanka in the final qualifying match here last week, recovered to hold their own after a shaky start against the spot-on Australian bowling.
Wavell Hinds, opening the batting in the absence of Shivnarine Chanderpaul, who was laid low with food poisoning, edged Nathan Bracken to the slips in the fourth over.
New man Dwayne Smith made eight when he flicked Lee to square-leg where Brad Hogg took a low catch to reduce the West Indies to 25-2.
Chris Gayle, who appeared set while striking four boundaries in his 24, edged a rising ball in Shane Watson's first over to wicket-keeper Adam Gilchrist.
Australian captain Ricky Ponting brought Michael Clarke on for one over and the young left-spinner responded by trapping Ramnaresh Sarwan leg-before with his sixth delivery.
But Morton and Lara hit back with their century stand that left Australia chasing a challenging 4.7 runs an over under lights on a pitch that could become slower.
Fall of wickets: 1-10 (Hinds), 2-25 (Smith), 3-47 (Gayle), 4-63 (Sarwan), 5-200 (Lara), 6-233 (Baugh)
Toss: West Indies
Umpires: Mark Benson (ENG) and Rudi Koertzen (RSA)
TV umpire: Aleem Dar (PAK)
Match referee: Mike Procter (RSA)