The decision has been made due to a "chronic shortage of accommodation" in Wales' four existing dispersal areas.
According to the latest UK Government figures, Cardiff, Swansea and Newport are all in the top ten of areas with the highest number of asylum seekers per head of population.
Conwy council said it "wants to play its role in assisting in the global humanitarian crisis."
The Home Office confirmed it was working with the authority, adding: "The UK has a proud history of granting asylum to those who need our protection."
In March 2016, the Home Office wrote to all 18 Welsh local authorities who do not house asylum seekers to ask them to start taking people in.
The Welsh Local Government Association (WLGA) said other local authorities in Wales were "in discussions" - but Conwy was the first to formally agree.
It said it welcomed Conwy's step, saying Welsh councils had "responded positively to the call to resettle Syrian refugees".
The Home Office wants every local authority in the UK to house more asylum seekers.
Although the scheme is currently voluntary, the Home Office said it will become mandatory if not enough councils come forward.
Conwy council said the aim of the pilot would be to "investigate the feasibility of Conwy becoming a asylum dispersal area", and would look at the "logistical and resource implications and community impact."
It added: "Conwy wants to play its role in assisting in the global humanitarian crisis.
"The Home Office has consulted its housing providers in Wales, and have concluded that the current geography of asylum dispersal needs to be widened."
Asylum seeker accommodation provider Clearsprings will provide one house in Conwy for "small numbers" of asylum seekers.
The council will decide on the location, size and household type of the accommodation. The pilot will be evaluated before any further commitment.
In March 2016, Cardiff had one asylum seeker per 263 people, Swansea one per 272, and Newport one per 302.
The dispersal scheme is separate to the Home Office's programme to resettle Syrian refugees, which saw around 300 re-homed across all Welsh local authorities by the end of last year.

Former England batsman Carberry hit 50 off 36 balls as Perth made 144-7 despite a maiden in the final over.
Pietersen, who fired a stunning six with a switch hit, and Luke Wright both made 31 but the visiting side were all out for 126 in the final over.
Perth will play Brett Lee's Sydney Sixers in Wednesday's final.
That match in Canberra will be the Australia paceman's final competitive game before his retirement.
There were boos ringing out from a vociferous home contingent when their tailender Andrew Tye swung and missed at six nearly identical slower deliveries from Clint McKay at the end of the Perth innings.
Pietersen looked to be guiding his team to victory but with 57 needed from 32 balls became the fifth wicket to fall when he advanced down the pitch to paceman Nathan Coulter-Nile and edged to the wicketkeeper.
Tye atoned for his batting with superb slower balls of his own to return 4-15 from his four overs and take Perth through to their fourth successive final as the Stars, who had won five matches in a row in this campaign, lost their fourth BBL semi-final.
On Saturday, Australia T20 left-hander Nic Maddison inspired the Sydney Sixers to a comprehensive 87-run win over the Adelaide Strikers in the first semi-final, with seven fours and five sixes in his 85 from 48 balls.
Sydney made 181-4 after choosing to bat and then dismissed Adelaide for 94 inside 15 overs, with burly left-arm seamer Doug Bollinger taking 3-21 and Sean Abbott claiming 2-14.
The Sixers also have former England opener Michael Lumb, and Carberry's former Hampshire team-mate, in their side.
Conwy county has agreed to house asylum seekers as part of a new pilot scheme.
[NEXT_CONCEPT]
Michael Carberry helped Perth Scorchers beat Kevin Pietersen's Melbourne Stars by 18 runs at the Waca and reach the Australian Big Bash League final.