Du Plessis was 69 not out and opener Elgar earlier made 61 to take South Africa to 201-4 by stumps on the second day, 55 behind Zimbabwe's first-innings score of 256.
South Africa had to work hard against the lowly-ranked Zimbabweans on a dry, slow pitch where batsmen found it hard to score freely through the opening two days.
Du Plessis ground out his 69 off 192 balls with eight fours and South Africa scored at less than 2 ½ runs per over in its first innings.
Nyumbu removed opener Alviro Petersen (32) and then the dangerous AB de Villiers for seven to have Zimbabwe's neighbor 157-4 midway through the day and in a little trouble.
Du Plessis combined for an unbroken 44-run stand with wicketkeeper-batsman Quinton de Kock to take South Africa past 200 and within sight of Zimbabwe. South Africa needs to win to retain the No. 1 ranking it won back from Australia with a series victory in Sri Lanka last month.
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