"He was born with just two settings - full throttle, then sleep," Bush said. "To us, his was the brightest of a thousand points of light," he added in reference to his father's signature call to volunteerism.
"He showed me what it means to be a President who serves with integrity, leads with courage and acts with love in his heart for the citizens of our country.
"When the history books are written, they will say that George HW Bush was a great president of the United States."
His voice cracked with emotion as he concluded his eulogy.
"Through our tears, let us know the blessings of knowing and loving you. A great and noble man, the best father a son or daughter could have."
Bush's eulogy followed an uplifting performance by Irish tenor Ronan Tynan - a friend of Bush who sang to the president in his dying hours.
Wednesday's state funeral caps a day-long homage that saw Bush lie in state for two days in the US Capitol rotunda.
Since Bush's death, Trump has traded his usual provocative posture for one of respect and solemnity, tweeting before heading to the cathedral about "a day of celebration for a great man."
But while the service allowed Washington to hit pause on the toxic rhetoric that has engulfed the country's politics, Trump and his Democratic predecessors appeared locked in an uneasy truce as it got underway.
Trump arrived and promptly shook hands with Obama and former first lady Michelle Obama. But his greetings stopped there, as he failed to acknowledge Hillary Clinton, his defeated Democratic rival in 2016.
Clinton sat stone faced, looking straight ahead, and the pair did not make eye contact.
Other dignitaries in the cathedral included Britain's Prince Charles, German Chancellor Angela Merkel, former Polish president Lech Walesa, former vice presidents Dan Quayle, Dick Cheney, Al Gore and Joe Biden, and former secretaries of state James Baker, Colin Powell and Condoleezza Rice.
Wednesday was a day of precision, patriotic ritual, and ceremony. George W. Bush and wife Laura stood stoically outside the US Capitol as an honor guard carried his father's flag-draped casket to the hearse.
In a show of respect, Pennsylvania Avenue was lined with well-wishers as the cortege proceeded toward the Neo-Gothic cathedral in the first presidential funeral since Gerald Ford died in late 2006.
Earlier, tens of thousands of Americans had quietly filed in to the Capitol to pay their respects to a man who steered the nation through turbulent times including the end of the Cold War -- and in a style dramatically different to the combative current president.
Trump's ascendancy to the head of the Republican Party saw him exchange vitriolic attacks with the Bushes, notably slamming the presidential son's 2003 invasion of Iraq as "one of the worst decisions in the history of our country."
But he has taken pains to demonstrate unity since Bush's death, and made a low-key visit to the Capitol late Monday with First Lady Melania Trump to salute Bush's casket.
'Civility and good will'
Trump declared Wednesday a national day of mourning. Many federal offices are closed along with Wall Street stock markets.
Congress also suspended votes -- even in the midst of a looming potential shutdown that requires congressional action before midnight Friday.
Bush was a decorated World War II aviator who nearly lost his life when he was shot down on a bombing mission.
He served as a congressman, envoy to China and ambassador to the United Nations, director of the Central Intelligence Agency, and vice president to Ronald Reagan before winning the White House.
At a time of extraordinary and deep political fissures, Bush was looked to this week as a gracious servant of country who aimed to bridge the political divide.
Bush "was America's last great soldier statesman," presidential historian Jon Meacham told the guests.
"His life code, as he said, was tell the truth, don't blame people, be strong, do your best, try hard, forgive, stay the course."
After the Washington service, Bush's casket will be flown back to Houston. The former head of state will lie in repose at St Martin's Episcopal Church, where the Bushes worshipped for decades, until Thursday's funeral.
He will be interred at the George H.W. Bush Presidential Library Center in College Station, Texas, next to his wife, who died in April, and their daughter Robin who died of leukemia at age three.