Suu Kyi and the National League for Democracy (NLD) swept to power in 2015 in a landslide victory ending decades of military rule.
But her tenure has been marred by a failure to speak up for Rohingya Muslims driven out of the country by the army and stumbling peace talks with insurgent groups in lawless border areas.
A mere 13 positions are in play in the country's second by-election since the national poll three years ago, but they are spread out across the country and include parliamentary and regional assembly seats.
Some two-dozen parties are in the mix and 69 candidates are taking part.
At one polling station in Yangon's Tamwe township residents showed support for Suu Kyi while acknowledging some of the criticism.
"I voted NLD this morning," Maung Maung, 34, a software engineer who lives in Tamwe, told AFP.
"I was a strong supporter of the NLD for years but during the years when NLD took power, there were some failures that they are working on," he added, without going into detail.
Aye Soe, a 52-year-old street vendor, expressed full-throated backing.
"I will support her until I die," she said.
Initial results are expected to be announced on Sunday.
Nobel laureate Suu Kyi's reputation at home is more secure than it is abroad, where her image as a rights icon has been shattered by the Rohingya crisis.
More than 720,000 from the stateless Muslim minority have fled to Bangladesh since a military crackdown in August 2017.
Huddled in crowded camps, they have recounted stories of murder, rape and villages burned to the ground.
Myanmar has denied almost all of the allegations, saying soldiers were defending themselves against Rohingya militants.
UN investigators have called for the situation to be referred to the International Criminal Court and for senior members of the Tatmadaw, as the armed forces are known, to be investigated on genocide charges.