For Cruz, the poll represents a massive 21-point jump from a survey conducted by the two outlets in October.
Cruz and Trump will face off Tuesday in Las Vegas at the next Republican presidential debate, sponsored by CNN.
"We have more work to do, but we can definitively say the message is working," Cruz's Iowa state director, Bryan English, told CNN in a statement.
The results are the most tangible sign yet that Cruz's strategy in Iowa -- heavily dependent on winning over pastors and their evangelical congregations -- is succeeding. For months, Cruz labored behind the scenes, courting and organizing influential faith leaders as he slowly worked to peel off the born-again believers of candidates like Ben Carson.
On Thursday -- after the poll was fielded -- the Texas senator won his biggest Christian booster yet: Bob Vander Plaats, who has endorsed the last two Iowa winners.
Ben Carson dropped to third place with 13%. Marco Rubio has 10%, which puts him in fourth place -- the same spot he held in October. Jeb Bush has 6% -- a 1% increase from two months ago.
Rand Paul, Mike Huckabee and Chris Christie are tied with 3%. The rest of the field, including Carly Fiorina and John Kasich, is at 2% or less.
The poll of 400 likely Republican caucus-goers was conducted between December 7 and 10, after Trump called for a ban on allowing Muslim immigrants into the U.S. It has a margin of error of plus or minus 4.9 percentage points.
Trump has often derided the Des Moines Register, Iowa's largest newspaper, as biased against him. He blasted the poll in a tweet Saturday night, though he appears to have mistakenly said the paper was biased "towards" him instead of against his candidacy.
"New CNN Iowa poll --- Trump 33, Cruz 20. Everyone else way down! Don't trust Des Moines Register poll - biased towards Trump!" he tweeted.
A CNN/ORC poll released on Monday found that the real estate mogul had 33% support among likely GOP caucus-goers in Iowa
, followed by Cruz at 20%. Nationally, a CNN/ORC poll released earlier this month
found the brash billionaire sitting atop the GOP field with 36%, 20 points ahead of Cruz, the GOP's runner-up.
Cruz now will have to contend with expectations that, for months, he has sought to carefully manage. Drawing a contrast with less well-resourced candidates, Cruz often told reporters that he did not view Iowa as a "must win," stressing that he could run a nationally-focused campaign.
Given his standing 50 days before the caucuses, that's a harder argument to make.
But much can change during the next seven weeks, for better or for worse. Cruz and his allies have so far spent very little money on advertising in Iowa, raising the prospect that his numbers could rise even more. Yet his opponents, for the most part, are only beginning to target him, attacks that when launched will make it difficult for Cruz to maintain his field-high favorability ratings -- 73% favorable, in this survey -- as voting draws closer.
A historic surge
In a story accompanying the poll results, the Register said no one had made Cruz's 21-point leap in five previous caucus cycles.
In the 2012 race, Mitt Romney, the GOP's eventual nominee, jumped eight points from 16% to 24% in the final poll before the vote. That came at the expense of Newt Gingrich, who plunged 13 points (25% to 12%) in the same survey.
In 2008, Mike Huckabee -- who would go on to capture the caucuses that year -- zoomed 17 points between October and November. In 2004, John Edwards also completed a 17-point leap in the final poll before the Democratic caucuses, while John Kerry rose 10 points, 15% to 25%. Kerry ended up defeating Edwards by 6 points in that year's caucuses
and eventually captured the party's nomination.
CORRECTION: A previous version of this story misstated the jump in Cruz's poll numbers since October.