The Russian, who won her two grand slam titles five years apart and hence believes this could be her year again following her 2009 success at Roland Garros, appeared to be heading for a straightforward victory when she led 3-0 in the second set.
But her biggest weakness has been her inconsistency and so it proved again on Monday when she allowed her left-handed rival to draw level at 4-4.
But with the sun setting over the Court One bullring, Kuznetsova let out a roar of delight as Safarova dragged a backhand into the net on match point.
The Russian will next meet in-form Romanian fourth seed Simona Halep for a place in the last four.
(Reporting by Pritha Sarkar, editing by Mark Meadows)
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