Like Bush, Dole is a military veteran. An aide helped Dole to his feet and supported him as he raised his left hand in salute to the flag-draped casket.
During Dole's salute to Bush, the attending crowd remained silent.
Bush died Friday at age 94.
Bush will lie in state until Wednesday morning. In the second row are Vice President Mike Pence, and his wife Karen Pence, former Vice President Dan Quayle, and his wife Marilyn Quayle and former Vice President Dick Cheney and his wife Lynn Cheney, former Vice President Joe Biden and his wife Jill Biden.
The late Mr Bush called Mr Trump a "blowhard".
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Mr Trump once mocked Mr Bush Snr's well-known "thousand points of light" phrase, used to describe the positive impact volunteers have on the country, asking "what the hell is that?", and was fiercely critical of both the former president's sons. He was a familiar presence in Washington: a congressman in the late 1960s, then Central Intelligence Agency director, envoy to China and Republican Party chairman. Bush's Instagram account. She was invited to the White House by first lady Melania Trump.
"Here lies a great man", said Rep. Paul Ryan, the House speaker, and "a gentle soul".
Although Bush's funeral services are suffused with the flourishes accorded presidents, by his choice they will not include a formal funeral procession through downtown Washington.
At 12:30, his body will be taken from the National Cathedral to Joint Base Andrews in Maryland to be flown back to Ellington Field in Houston where final ceremonies will take place at St. Martin's Episcopal Church before his internment at his presidential library in College Park, Texas.