The former Aberconwy assembly member beat independent Goronwy Edwards by one vote, but no one party has a big enough majority to secure overall control.
Mr Jones said he wanted to involve all the political groups on the council in the running of the authority.
He said the aftermath of the election was about "spending money wisely" and "not about politics".
"Coalitions shunt others to a position outside the ruling body. It's not right to keep any group out of office," Mr Jones said.
"We need to develop an inclusive style of administration which respects what the people of Conwy voted for."
Mr Jones later admitted it was possible not all political parties would be able to take part in an administration, but said the offer would be extended to all parties.
Previous leader Dilwyn Roberts stepped down and retired as a councillor before May's local elections.
Conwy council has elected Plaid Cymru's Gareth Jones as its leader.