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Bournemouth took control early on after Marcus Tavernier opened the scoring, before Newcastle fell further behind when defender Malick Thiaw inadvertently turned the ball into his own net.
Newcastle responded through Harvey Barnes, who pulled one back with a powerful strike in the 37th minute.
Substitute Jacob Ramsey then completed the comeback in the closing stages, scoring the equaliser to rescue Newcastle from a home defeat.
The match also marked the Newcastle debut of French midfielder Matthis Fernandez-Pardo, who came on as a second-half substitute after joining the club in the summer transfer window.
The draw left Newcastle five points from three matches. They opened their campaign with a draw against Liverpool, followed by a win over Tottenham, before sharing the points with Bournemouth.