It looked like the Rebels wouldn't cross the stripe in the first half when five-eighth Jack Debreczeni was held up, but Higginbotham got his first in the final minute as they sent the ball wide and he shrugged off the tackle of Cronje.
With Debreczeni adding the extras the margin was 10-7 at the break, with the error count from both sides in double figures.
The Sharks struck a double blow early in the second stanza when electric fullback Lwazi Mvovo raced 70m after an intercept and then eight minutes later Heimar William cut back inside through a gap to open up a 22-7 lead.
The Rebels kept fighting with Higginbotham, who holds the Super Rugby record for the most tries scored by a forward, leading the charge.
He carried two defenders over the line and then replacement back Bryce Hegarty scored with five minutes left to give the visitors a real sniff.
But they were unable to get the necessary fourth try and leave South Africa with just a bonus point from two games.
-AAP