The unidentified foreigner, who is being held in military custody at an undisclosed location, was seized during a Saturday morning raid on a flat on the eastern outskirts of Bangkok.
Investigators say he was found with bomb-making equipment and dozens of fake passports.
National police spokesman Prawut Thavornsiri said officers believed the suspect was part of a crime group who helped illegal migrants obtain counterfeit documents -- and that the bomb attack was retaliation for a recent crackdown by Thai authorities.
"They (the gang) are unsatisfied with police arresting illegal entrants," he told Channel 3 in a telephone interview without elaborating how investigators knew this.
"It's a network that fakes nationalities and sends them (illegal migrants) on to third countries," he added.
The blast that hit the Erawan shrine in a busy Bangkok shopping district on August 17 was Thailand's worst single mass-casualty attack, killing 20 people -- most of them ethnic Chinese tourists from across Asia.
Thai authorities have played down any suggestion the attack was launched by international terrorists or specifically targeted Chinese tourists.
Authorities have yet to say what nationality the detained man is but they believe he had accomplices for whom they are now searching.
Outside police headquarters Prawut told reporters that investigators were sifting through more than 1,000 phone numbers as well as hundreds of passport pages to track the gang, while DNA samples had also been taken from the suspect.
Investigators are also working "with several embassies" to ascertain the man's identity as well as using multiple translators, he added.
Asked which languages, Prawut confirmed English but would not be drawn on the others.
Defence Minister Prawit Wongsuwan also told AFP the suspect is known to speak some English.
However the suspect appeared to be refusing to cooperate with investigators.
"The interrogation is not making progress because the suspect is not really giving useful information," army chief General Udomdej Sitabutr told AFP.
"We have to conduct further interrogations and make him better understand so he will be more cooperative -- while we have to be careful not to violate the suspect's rights," he added.
On Sunday afternoon police searched a flat in a northeast Bangkok suburb near the previous day's raid but no arrests were made, an AFP reporter on the scene said.
Policemen conduct a search operation at apartment blocks on the outskirts of Bangkok on August 30, 2015. - AFP PHOTO / Christophe ARCHAMBAULT
Forgery and smuggling hub
Bangkok's crime groups have long had a reputation for producing counterfeit documents, while Thailand has been a major regional hub for both people-smuggling and people-trafficking.
Police have made no major recent announcements about busts against counterfeiters.
Earlier this year they cracked down on the regional people-trafficking trade. But that trade centres around impoverished Bangladeshi and Myanmar migrants, usually held in jungle camps and on boats -- a demographic unlikely to be rich enough to buy forged passports.
Until the recent arrest the investigation into the Erawan shrine bomber had appeared unsure and at times erratic.
Further confusion mounted late Saturday when an erroneous picture of a suicide bomb vest appeared in a nationally televised junta broadcast alongside other items found in the arrested man's flat.
On Sunday the junta said the picture should not have been included and blamed Thai media for adding it, although it is not clear whether broadcasters could have altered an official junta address which ran on all free channels.
The manhunt has focused on a prime suspect, described as a foreign man, who was captured on security footage wearing a yellow T-shirt and leaving a bag at the shrine moments before the blast.
Authorities have not yet said whether they believe the suspect now detained is the same as the man in the video footage.
Mystery has surrounded the unprecedented bomb blast, for which no group has claimed responsibility.
Potential perpetrators named by the police and experts have included international jihadists, members of Thailand's southern Malay-Muslim insurgency, militants on both sides of the country's festering political divide or someone with a personal grudge.
Speculation has grown over involvement by China's ethnic Uighur Muslim minority -- or their co-religious sympathisers -- following Thailand's forced repatriation of more than 100 Uighur refugees last month to an uncertain fate in China.
The sudden repatriation of the Turkic-speaking group, among dozens detained in the kingdom for illegal entry last year after presenting themselves to police as Turkish, triggered fury in Istanbul with Bangkok's consulate there stormed by protesters.
But Defence Minister Prawit called on both the media and Thais to be patient.
"Don't talk about Turkish or not Turkish," he told AFP. "We have to investigate."
