Mirziyoyev became interim president following Karimov's death from a stroke in September at the age of 78.
The Central Electoral Commission said around 88 percent of the 20 million-plus electorate voted in a ballot that ended at 1500 GMT on Sunday.
Preliminary results are due to be released at 1100 GMT.
Mirziyoyev, who was appointed prime minister in 2003, became interim president following Karimov's death from a stroke in September at the age of 78.
Representing the same Liberal-Democratic Party that Karimov stood for in the last presidential vote in 2015, Mirziyoyev faced three other challengers in a bid to secure a five-year term.
But analysts noted that the other candidates were not critical of Mirziyoyev or of the authoritarian regime in the former Soviet republic which borders Afghanistan and where Beijing, Moscow, and Washington all vie for influence.
"The format for Uzbek elections has not changed since Karimov's death because the regime has not had time to think of anything different," said Kamoliddin Rabbimov, an Uzbek political analyst who lives in France.
"Uzbekistan has its own specific take on democracy. There will be no surprises here," he added.
Karimov's 27-year reign began in 1989 at the tail-end of the Soviet era and was often criticised for extreme abuses of human rights.
In the run-up to Sunday's poll, Mirziyoyev, 59, created an online forum for public complaints and pledged to prioritise economic reforms as many Uzbeks struggle to eke out a living.
The election commission published a series of glowingly optimistic comments attributed to voters about the poll.
"I realised the importance of my voice in the future development of the country which made me proud," student Sevara Foziljonova was quoted as saying on the commission website.
"There was a time when we had to agree with everything," the CEC quoted 106-year-old World War II veteran Badal-bobo Khuramov as saying, referring to the Soviet era.
"Now everything has changed," said the man the commission described as a grandfather to "around 50 grandchildren".
As premier for 13 years, Mirziyoyev was regularly touted as a potential successor to Karimov, along with current deputy prime minister Rustam Azimov and Karimov's eldest daughter Gulnara Karimova.
But businesswoman-cum-popstar Karimova, 44, has been reportedly under house arrest since 2014 after publicly feuding with her mother and her younger sister Lola.
She did not attend her father's funeral, and neither did her two children, with her eldest son Islam Karimov Jr, who lives in London, calling on authorities to prove she was alive and well in a recent interview with the BBC.
Karimova is also under investigation in connection with a corruption probe involving Western telecoms firms that US and European investigators say paid her billions of dollars in bribes to secure access to the national market.
Under Karimov, Uzbekistan enjoyed mostly cordial relations with foreign powers active in the region but kept all of them at arm's length while regularly threatening smaller neighbours Kyrgyzstan and Tajikistan.
In 2005, it ejected US forces from a military base used for Afghanistan operations over Washington's human rights criticisms.
And in 2012, it exited a Moscow-led security bloc.
After Karimov's death, Mirziyoyev was quick to meet with Russian leader Vladimir Putin and last month hosted President Recep Tayyip Erdogan of Turkey, with whom Karimov had patchy relations.
"Under Karimov, Uzbekistan withdrew into itself, to its great cost, and this may be the main difference under Mirziyoyev," said analyst Rabbimov.
"The policy of not joining military blocs and hosting bases may remain, but I think Uzbekistan will begin to participate in more international initiatives.
"He will see the country can't survive in isolation."