Dewhurst is seeking a fourth term and has said it would most likely be his last. Patrick has been in the Senate since 2007.
Patrick was leading Dewhurst in election results late Tuesday night.
Those two emerged as the top candidates from a bitter campaign that also included state Land Commissioner Jerry Patterson and Agriculture Commissioner Todd Staples.
Each has claimed to be the most conservative and heavily courted tea party groups that were seen as the key to Dewhurst's defeat in a 2012 campaign for U.S. Senate.
The runoff election has been scheduled for May 27.
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