Mr Cruz, a conservative US senator from Texas, led the billionaire businessman by 27.7% to 24.3%, with 98% of precincts reporting results in the Midwestern US state.
Florida Senator Marco Rubio, viewed by many Republicans as a more mainstream alternative, was nipping at Mr Trump's heels in third place at 23.1%.
Mrs Clinton, a former Secretary of State, has declared victory despite a wafer-thin lead of 49.8% over 49.6% for Mr Sanders, a US senator from Vermont, with 94% of precincts reporting in that race.
Delivering his victory speech, Mr Cruz told supporters: "Tonight is a victory for courageous conservatives across Iowa and across this great nation."
Mr Trump, toning down his trademark bombast, told fans: "We finished second and I just want to say, I'm really honoured and I want to congratulate Ted."
But the real estate billionaire, who had never stood for election before, went on to predict he would still win the Republican nomination.
The third Democratic candidate, Martin O'Malley, has suspended his campaign after trailing badly. He has gained less than 1% of Iowa's votes.
Republican Mike Huckabee has also dropped out of the race.
Turnout has been huge on both sides for the first-in-the-nation contest to pick candidates for November's US White House elections.
Long lines were reported at many caucus sites with many new voter registrations as Iowans gathered in schools, libraries and private homes.
At one Des Moines precinct, Post-it notes were used after ballot papers ran out. At another venue, three times the expected number of voters turned up.
It is the moment of truth for the 12 Republicans and three Democrats in the race after months of rallies, televised debates and tens of millions of dollars of political advertising.
Mr Sanders, the 74-year-old self-described democratic socialist who has electrified youthful crowds, urged voters to help him "make history" with a win in Iowa.
Mrs Clinton, the former First Lady, was hoping the imprimatur of the party establishment a muscular get-out-the-vote operation would seal victory.
A loss in Iowa for Mrs Clinton, who aims to be the first female US president, would be a stinging replay of her 2008 defeat to another upstart challenger, now-President Barack Obama.
Based on the outcome, candidates will be awarded delegates to the parties' national conventions in July when the presidential nominee is crowned.
A strong showing in the Midwestern state could generate momentum into next week's New Hampshire primary and Super Tuesday on 1 March, while also-rans will face pressure to drop out.
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