It was more-or-less a draw.
Neither army was destroyed, though the casualities were high.
Eventually the Confederates (Bragg) withdrew, so it was chalked-up as a Northern victory under Rosecrans.
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The North won several battles during the Civil War. Three of the battles won were the Battle of Murfreesboro, Battle of Vicksburg, and the Battle of Gettysburg.
The Union General, William Rosecrans
During the Civil War, a large-scale confrontation at Murfreesboro between Union and Confederate armies, which has come to be known as the Second Battle of Murfreesboro (or, the Battle of Stones River), was won by the Union troops led by Major General William S. Rosecrans. Taking place from December 31, 1862, through January 2, 1863, the battle was won by Rosecrans for several reasons: his force was superior in the numbers of artillery weapons, his sub-commanders proved themselves to be very cool in the face of repeated Confederate attacks, and the imminent arrival of fresh reinforcements convinced the Confederates that retreat was necessary.
It's what they said about Murfreesboro (Stone's River), Tennessee.
The Second Battle of Murfeesboro (as named by the Confederates) was called the Battle of Stones River by the Union forces who named battles after the nearest stream or river.