Prakash secured 31 votes, while Soren, a former chief minister, won 30.
Congress candidate Shahzada Anwar finished third by bagging 18 votes, the official said.
This will be Soren's third term as a Rajya Sabha member. Soren, whose Jharkhand Mukti Morcha (JMM) now governs Jharkhand, had also served as a Union minister.
It will be Prakash's maiden term in the Upper House of Parliament.
Polling in the two seats was held during the day as the tenure of Independent member Parimal Nathwani and RJD's Prem Chand Gupta ended in April.