The visitors will take a commanding 271-run lead into the fourth day after reaching 139 for two in their second innings.
Off-spinner Botha dismantled Pakistan’s middle order by picking up three quick wickets and Morkel completed a bad day for the designated hosts as he grabbed five for 54.
Pakistan lost their last five wickets for the addition of just 28 runs and face an uphill task to save the test.
Azhar Ali was the only Pakistan batsman to defy South Africa for any length of time as he ground his way to 56 off 172 balls.
He was eventually dismissed when he fended a Morkel short ball to Hashim Amla at short leg as Pakistan’s innings fell away in the face of some disconcerting bounce that Morkel extracted off a lifeless pitch.
Paceman Dale Steyn sparked the collapse after lunch when he had Adnan Akmal (10) caught behind before Botha and Morkel took centre stage.
Off-spinner Botha found both bounce and turn to trouble the batsmen and claimed the scalps of Younus Khan (35), Misbah-ul-Haq (nine) and Umar Akmal (four) to bag three wickets for 61.
Morkel had Rehman (one) caught at third slip before he dismissed Ali.
Morkel then ended the innings in the space of five balls when he had both Wahab Riaz (five) and Saeed Ajmal (two) caught by keeper Mark Boucher.
South Africa lost Alviro Petersen (26) and Graeme Smith (34) before an unbroken third-wicket stand of 63 between Amla (44 not out) and Jacques Kallis (32 not out) guided them to the close of play.
