"I offer my apology for having been unable to prevent this accident from happening and unable to properly respond to it afterwards," he said.
"I believed I, as the prime minister, certainly had to take responsibility and resign."
Parents and relatives of the missing and the dead have blasted the response to the sinking, saying delays in launching the rescue had cost lives.
There has also been rage over perceived corruption and lax safety standards that may have led to the disaster, with claims that the ferry was overloaded and that the passenger list was inaccurate and incomplete.
"Looking at the latest accident I came to a painful realisation that there is too much deep-rooted evil and corruption in our society," Chung said.
The role of prime minister is largely ceremonial in South Korea, with the lion's share of executive power vested in the presidency of Park Geun-Hye.
Park has decided to accept Chung's resignation but only after the government finishes the entire rescue and salvage operation, her spokesman said. In Jindo, the nearest island to the wreck, relatives of the dead and missing were unimpressed by Chung's move.
"So what?" snapped Ji Hyung-Soo. "My son is there in the sea. His resignation will never ease my bitterness and sadness. "Anybody responsible for this disaster must be punished severely, but the most urgent thing to do now is to recover the bodies as soon as possible."
Prosecutors looking to mete out the justice sought by relatives such as Ji on Sunday raided the offices of state sea traffic controllers in Jeju island, the ferry's intended destination, and in Jindo.
They seized records of radio communication with the Sewol and surveillance video footage, Yonhap news agency said.
A transcript of conversations released earlier revealed panic and indecision among crew and sea traffic controllers in the crucial final moments, with neither able to make the call to evacuate passengers. The confirmed death toll from the tragedy stood Sunday at 188, barely changed in two days. A total of 114 people were still unaccounted for, with many bodies believed trapped in the sunken vessel.