Mr Cruz, a conservative US senator from Texas, led the billionaire businessman by 27.7% to 24.4%, with 90% 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%.
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 90% of precincts reporting in that race.
Toning down his trademark bombast, Mr Trump told supporters: "We finished second and I just want to say, I'm really honoured and I want to congratulate Ted."
But he 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.
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, is urging voters to help him "make history" with a win in Iowa.
Mrs Clinton, the former First Lady, is hoping the imprimatur of the party establishment a muscular get-out-the-vote operation will 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.
"I know how to do this and I'm ready," she told CNN hours before voting.
Mr Trump, a real estate billionaire who has never stood for election before, held two rallies on Monday as he strived to fire up supporters of his plainspoken populism.
Iowa looked like a must-win for Mr Cruz, who visited all of the state's 99 counties and courted influential evangelical and conservative leaders.
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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