Smith and Amla brought up their 50-run stand off 83 balls before facing a total of 17 overs in constructing their second-wicket partnership.
Smith was struck on the helmet by a vicious short ball from paceman Doug Bracewell early in the day but South Africa's captain battled his way to 40 not out off 73 deliveries at the break.
Amla, after a relatively slow start to his innings, began to play a few more shots shortly before lunch and he reached the interval on 32 off 50 balls with five fours.
South Africa won the toss and elected to bat on a slowish pitch which offered up a touch of movement for the seam bowlers with New Zealand including debutant all-rounder Colin Munro in their starting XI.
Smith and Alviro Petersen combined for a 29-run opening stand with Petersen doing the bulk of the scoring with 21 off 40 balls.
The right-handed Petersen departed when he top-edged an attempted hook shot off the bowling of Bracewell and was caught by Jeetan Patel at fine leg.
Bracewell ended the session with figures of one for 18 off eight overs. (Writing by Jason Humphries in Durban; Editing by Clare Fallon)