Cruz has the backing of 28% of Republican voters nationwide, unseating Trump, who won the support of 26% in the latest NBC News/Wall Street Journal poll.
But Cruz's 2-point edge is within the poll's margin of error, and it's not clear if the survey captures real movement in the race or is simply an outlier.
Florida Sen. Marco Rubio comes in third with 17% support, followed by Ohio Gov. John Kasich at 11%, Ben Carson at 10% and former Florida Gov. Jeb Bush in last place with 4%.
The results are a major change from last month's NBC News/Wall Street Journal poll
, when Trump held a 13-point lead over Cruz, 33% to 20%.
But Trump dismissed the new results, when asked about them Wednesday night during a town hall hosted by MSNBC.
"I have never done well in the Wall Street Journal poll. I think somebody at Wall Street Journal doesn't like me but I never do well with the Wall Street Journal poll," Trump said. "So I don't know. They do these small samples and I don't know exactly what it represents."
Cruz, however, crowed about his standing in the race Wednesday.
"The sound you're hearing is the sound of screams coming from Washington, D.C.," Cruz told supporters at a rally in Spartanburg, South Carolina, on Wednesday afternoon. "But what's happening nationally is indicative of the stakes in this race."
The latest survey was conducted by Republican pollster Bill McInturff and Democratic pollster Peter Hart after Saturday's GOP debate in South Carolina.
McInturff told NBC News it was too early to tell whether the change at the front of the Republican field was permanent or only a temporary "pause."
"When you see a number this different, it means you might be right on top of a shift in the campaign," McInturff said. "What you don't know yet is if the change is going to take place or if it is a momentary 'pause' before the numbers snap back into place."
Trump has continued to dominate the next two Republican contests, however, according to a pair of CNN/ORC polls released this week. In South Carolina, Trump leads Cruz 38%-22%,
and in Nevada, Trump is beating Rubio 45%-19%
Another NBC/Wall Street Journal poll released Wednesday found registered voters to be divided on whether the Senate should vote this year on a replacement for late Supreme Court Justice Antonin Scalia.
Forty-three percent believe the Senate should vote on a replacement this year rather than wait for President Barack Obama's term to end, versus 42% who oppose a vote.
Along party lines, Democrats overwhelmingly support a vote this year, Republicans overwhelmingly reject a vote, and independents are split: 43% in favor and 42% opposed.
Obama has vowed to press forward and submit a nominee
to the Senate, despite promises from Senate Majority Leader Mitch McConnell that he would block any vote.
The NBC News/WSJ pollsters contacted 800 registered voters for the question on the Supreme Court, with a margin of error of +/- 3.5 percentage points, and 400 Republican primary voters for the Republican field, with a margin of error of 4.9 percentage points.