Preliminary results show he took 59% of the vote, while his centre-left opponent Sandra Torres won 41%.
Giammattei, making his fourth bid for the presidency, is the favorite in a CID-Gallup poll which surveyed 1,216 people between July 29 and August 5.
Noting that in July a U.S. federal judge in California had already blocked a Trump administration rule barring asylum applications at the U.S. -Mexico border, Giammattei said he was hopeful that Guatemala's own agreement with the U.S. might be set to change.
Three of the last four elected presidents have been arrested after leaving office on charges of graft, and Morales himself chose to disband and bar a United Nations -supported anti-corruption commission after he became a target for alleged campaign finance violations.
Voter turnout was reportedly low amid disillusionment with Mr Morales' administration and scepticism over corruption.
"It will be an enormous honour to be president of this country that I love so much".
Remittances from US-based Guatemalans reached a record $9.3 billion (€8.3 billion) previous year. "I have no words to say how grateful I am".
Under the agreement, migrants passing through Guatemala en route to the U.S. would have to apply for asylum in the former rather than in the US.
"The aim is fulfilled", Giammattei said earlier. "If we don't have the capacity to look after our own people, imagine what it will be like for foreigners".
"They will not find a president, who is away from the people".
On July 6, Morales's administration signed an agreement with the U.S. that would require Salvadorans and Hondurans to request asylum in Guatemala if they cross through the country to reach the US.
Alejandro Giammattei, presidential candidate with the Vamos party, talks to supporters at his campaign headquarters after the polls closed on election day in Guatemala City, August 11, 2019.
"I hope that during this transition the doors will open to get more information so we can see what, from a diplomatic point of view, we can do to remove from this deal the things that are not right for us", he said.
A 63-year-old who walks with crutches because of multiple sclerosis, his mild-mannered demeanor belies an inner strength forged through adversity.
Adding to his challenges, Fitch Ratings said the divided political landscape will make it harder for the president to reverse declining tax collection that the agency cited in April when it revised Guatemala's sovereign outlook to negative.
But rejecting the migration pact would run the "risk of retaliation from Trump", Grais-Targow said, after the USA leader threatened a travel ban, tariffs and remittance fees if Guatemala didn't bend to his will.
He said he would back a bill declaring gang members "terrorists" so that if imprisoned they would have no right to conjugal visits and would be forced to work in jail.
Giammattei will not take office until January, by which time Guatemala may be under severe pressure from the deal, which effectively turns the country into a buffer zone by forcing migrants to apply for asylum there rather than in the United States. Seven inmates died in the operation, which sought to reimpose control over a facility that had become notorious for criminal activities.
More than 250,000 Guatemalans were detained between October 2018 and July this year for trying to enter the U.S. illegally, Washington's embassy said.
In addition to migration, Guatemalans have said they are concerned about crime, unemployment, the rising costs of living and entrenched corruption.
Morales, barred by Guatemalan law from seeking a second term, urged his replacement to reduce undocumented migration, improve education and tackle chronic malnutrition in the under-fives, which affects 46% of infants.
Analysts do not expect Giammattei to reverse the decision by Morales to expel the UN-backed International Commission against Impunity in Guatemala, which helped to launch a massive anti-corruption drive and bring down President Otto Perez Molina in 2015.