Chigumbura made a battling 90 in his side's 171 all out, a mark which South Africa rattled off in just 27.2 overs for the loss of three wickets.
Quinton de Kock continued his rich run of form with 84 in 75 balls to lead the chase, but his efforts were at least matched by Chigumbura's backs-against-the-wall knock.
There were changes in both sides for what was a dead rubber in terms of the series result, the tourists resting skipper AB de Villiers, Hashim Amla and Imran Tahir in favour of Marchant de Lange and debutant pair Rilee Rossouw and Mthokozisi Shezi.
Zimbabwe also showed three changes, drafting in Malcolm Waller, Tafadzwa Kamungozi and Tinashe Panyangara.
Stand-in Proteas captain Faf du Plessis won the toss and inserted the hosts, who promptly lost openers Hamilton Masakadza and Richmond Mutumbami with just four on the scoreboard.
Chigumbura restored order alongside Sikandar Raza (30), sharing a 51-run stand before the latter was dismissed by Ryan McLaren.
Chigumbura, who hit 10 fours and two sixes in his 122-ball stay, next added 44 with Sean Williams but once the number five had departed for 17, Zimbabwe collapsed.
No other batsman reached double figures as a pacy De Lange took three for 31, including Chigumbura to close the innings, while Kyle Abbott and Wayne Parnell took two wickets apiece.
Shezi picked up number 10 Kamungozi for a maiden scalp.
His fellow newcomer Roussouw fared less well opening the South Africa reply, run out by Waller for duck, but it was a rare success for Zimbabwe.
De Kock and Du Plessis scored at a rapid rate before the captain was bowled by Williams for 40.
Man-of-the-series De Kock looked in no mood to slow down and was hot on the heels of a sixth one-day hundred when he slapped Panyangara straight to deep midwicket.
JP Duminy (28no) and David Miller (13no) finished the job, with the latter sealing victory with six off John Nyumbu.