The hosts controlled much of the match, leading 15-13 at halftime, and even extended their advantage to four goals in the second half.
However, Egypt faltered in the final ten minutes as Spain capitalized on their superior physical conditioning to level the score and clinch victory by a two-goal margin.
Egypt will now face Norway in the fifth-place play-off on Friday.
Earlier on Thursday, Sweden edged Norway 35-34 in a thrilling quarter-final and will meet Spain in the semi-finals.
The other semi-final will see Denmark, who defeated Iceland 32-30, take on Germany, who advanced with a 32-31 win over Hungary.
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