The British number three saved two match points before coming through by a 4-6 7-6 7-5 scoreline, but tensions spilled over during the second-set tie-break after Ostapenko appeared to throw her racquet towards a ball boy.
The Latvian claimed that it was accidental, saying that the racquet slipped from her hand, while Broady called for an immediate disqualification, before the court supervisor let Ostapenko off with a warning.
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Despite levelling the match at one set apiece, the incident affected Broady and Ostapenko regained control, but she produced a comeback, reeling off six successive games to set up a quarter-final contest with Sloane Stephens.
The conclusion of the match was also marred, with Broady and Ostapenko continuing to confront each other, but it was the Stockport hitter who will progress as she looks to confirm a career-high ranking when the standings are updated next Monday.