Neither player was able to record a significant break of serve during the opening set, which as a result headed into a tie-break.
That was also quite evenly contested until Italian Errani claimed a decisive mini-break to take the breaker 7-5.
She then looked to be heading for a routine victory when she took a 3-0 lead at the start of the second set.
However, world number six Kerber responded to win the next three games in succession, levelling up the set in the process.
Errani then held for a 4-3 lead, before claiming another break of Kerber's serve. She then held her composure to book her place in the final eight.