Chigumbura arrived at the crease in just the second over after Zimbabwe had crashed to 4 for two in seven deliveries, and was the last man out as the hosts were bowled out in 39.5 overs.
Fast bowler Marchant de Lange recorded figures of 3 for 31 as he returned from a long injury layoff to play his second ODI for South Africa more than two years after he played his first.
Chigumbura revived the innings alongside Sikandar Raza Butt, who made 30 from 24 balls, but after Raza was dismissed by Ryan McLaren and the in-form Sean Williams was bowled by Aaron Phangiso, the Zimbabweans lost five for 16 in six overs.
With Zimbabwe 119 for nine, Chigumbura went to his half-century in 102 deliveries, and then cashed in during the power play, hitting five fours and two sixes as he added 40 in 20 balls.
Abbott and Wayne Parnell finished with two wickets apiece, while debutant Mthokozisi Shezi recorded figures of 1 for 8 from six overs.
The poor Zimbabwean total came after the hosts had sprung a surprise by dropping premier batsman Brendan Taylor.