Wicketkeeper-batsman Klaasen will act as cover for Quinton de Kock, while De Bruyn is retained from the squad that beat Sri Lanka by an innings an 118 runs in their last test in Johannesburg in January.
Right-hander Klaasen finished the domestic four-day competition with 635 runs at an average of 48.84, including a best of 195.
Fast bowler Duanne Olivier, the leading wicket-taker in domestic cricket this season, made his debut in that game, posting match figures of 5-57, and has been retained.
There is also a recall for fit-again Chris Morris, whose last test was against England in January 2016, while Morkel has also spent the last 13 months on the sidelines with back problems.
"Heinrich Klaasen was one of the top performers throughout the (domestic) season and he is rewarded for his form," Cricket South Africa selection convener Linda Zondi said in a statement on Friday.
"Morne Morkel has worked hard on his fitness over the past few months, and we believe he is now 100 percent fit and fully able to compete for his place in the team."
The other seamers in the squad are Vernon Philander, Kagiso Rabada and Wayne Parnell, with Keshav Maharaj the only recognised spinner.
The side will be captained by Faf du Plessis, with leading batsmen AB de Villiers announcing earlier in the year that he would not be available for this tour.
The first test starts in Dunedin on March 8. South Africa have won their last three test series, away in Australia and home to New Zealand and Sri Lanka.
South Africa squad: Faf du Plessis (captain), Hashim Amla, Temba Bavuma, Stephen Cook, Theunis de Bruyn, Quinton de Kock, JP Duminy, Dean Elgar, Heinrich Klaasen, Keshav Maharaj, Morne Morkel, Chris Morris, Duanne Olivier, Wayne Parnell, Vernon Philander, Kagiso Rabada
(Reporting by Nick Said; Editing by Hugh Lawson)
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