The results are an improvement for Trump, who trailed Cruz 24 percent to 28 percent in the same poll less than three weeks ago.
The poll also shows Trump in a close race with Cruz among likely voters who are white evangelical Christians, 31 percent to 28 percent. The two candidates have been battling over the influential bloc of voters in the state, with each touting endorsements from evangelical heavyweights in recent days.
As he has throughout the campaign, Trump has a big advantage with non-college graduates, winning 42 percent of their support compared to 24 percent for Cruz and 12 percent for Rubio.
Among likely caucus-goers who are Republican college graduates, Trump receives 20 percent, while Cruz gets 26 percent and Rubio gets 24 percent.