Sheri Sam hit a three-pointer with 39 seconds remaining to give the Fever a 57-56 advantage and ensured the victory by intercepting Mwadi Mabika's pass with three seconds left.
The Sparks missed four potential go-ahead shots in the final 20 seconds.
Sam had nine points and Tammy Sutton-Brown added 14 on seven of 10 shooting for Indiana, which has won three of its last four games.
Sidney Spencer scored 18 points and grabbed seven rebounds and Sherill Baker added 15 points and six assists for Los Angeles, which shot a dismal 31 percent (20-of-65) from the field.
The Sparks lost their season-high fourth straight game.
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