They were found guilty on lesser obstruction of justice charges, relating in Salah's case to a statement denying membership in Hamas.
Ashqar was found guilty on two counts for refusing to answer questions from two grand juries.
The obstruction charges call for up to a five-year sentence, but also allow for probation, lawyers said.
"It is better than we thought," a tearful Salah, a businessman from the Chicago suburb of Bridgeview, Illinois, said as he hugged supporters just outside the courtroom.
"We are good people, not terrorists." Michael Deutsch, his lawyer, said: "It's a great victory." The verdicts will be appealed, Matthew Piers, defence attorney, said.
The judge set sentencing for June 15, and allowed both defendants to remain free on bond.
Women were on their knees outside the courtroom praying before and after the jury's verdict was read, which came following three weeks of jury deliberations and a 10-week trial.
The 2004 federal grand jury indictment said Salah, who became a US citizen in 1979, was the point person from 1988 until 2003 for money transfers that went to Hamas, and also provided recruits and delivered messages on behalf of Mousa Mohammed Abu Marzook, the accused Hamas leader in the US.
A former US doctoral student, Marzook is believed to be in Syria. Prosecutors described Ashqar, who has taught at Howard University in Washington, as a co-ordinator and archivist for Hamas's military wing.
Hamas leads the Palestinian government, but Israel, the European Union and the US regard it as a terrorist group and it is illegal for US citizens to contribute to it.
"This is the second time the United States government has tried to bring this [Israeli-Palestinian] conflict into a criminal court of law," Ashqar's attorney, William Moffitt, said.
He referred to a Tampa, Florida, jury's acquittal of a computer sciences professor on most terror funding charges in 2005.
In January 1993, Salah was arrested at an Israeli checkpoint and found to be carrying nearly $100,000 in cash.
His two-month interrogation produced written and tape-recorded confessions that helped send him to an Israeli prison for four and a half years.
Lawyers for both Salah and Ashqar said they were only involved in charitable work.