She said her party had to reach out to people who felt let down by politics and the assembly and she had "something unique to offer the people of Wales".
"As a party we have broken the mould today by electing a woman," she said.
"If you have been turned off by politics, by the way the Labour Party has let Wales down, or the Conservatives' attitude, or Plaid's abandonment of principles, then come.
"We will re-ignite the flame of liberalism that once burnt so bright in this country.
"I am determined as leader of this party that the Welsh Liberal Democrats will blaze a trail for a new politics in Wales."
Ms Williams also paid tribute to her "formidable opponent" Ms Randerson and called Mr German a "tremendous leader of this party".
Mrs Randerson said the contest had "ignited" interest in the party and "re-engaged" its membership.
Kirsty has a strong record on the issues that matter most to the people of Wales
Lib Dem leader Nick Clegg
Addressing the party's new leader she told Ms Williams: "You have a mandate for change within the party and a platform for bringing change to the nation."
"Kirsty, you have my deepest and sincere congratulations.
"I have worked with you since the day the assembly started. Your qualities and strengths are formidable.
"You have my word that from this moment onwards I will be with you every step of the way."
Liberal Democrat leader Nick Clegg was one of the first to congratulate his new Welsh counterpart on her success.
"Kirsty's election as leader is excellent news for the Welsh Liberal Democrats and for Wales," he said.
"I know that Kirsty will be a strong champion for liberal values in the Senedd, helping to make Wales a fairer and greener nation.
"Kirsty has a strong record on the issues that matter most to the people of Wales, taking the fight to both Labour and Plaid Cymru in their heartlands."
Labour leader and First Minister Rhodri Morgan added his own congratulations and took time to pay tribute to the departing Mr German.
KIRSTY WILLIAMS FACTFILE
Born: Taunton, Somerset
Previous occupation: Marketing executive
Constituency: AM for Brecon and Radnorshire since 1999
Supporters include: Mark Williams MP, AMs Mick Bates and Peter Black, Powys councillor Liam Fitzpatrick, Cardiff councillor Gareth Aubrey
"Not only is she the first woman to be elected as a party leader in Wales, she has already shown her abilities as a feisty campaigner back in the referendum campaign in 1997, and in the assembly over the past nine years," he said.
"Can I also take this opportunity to pay tribute to Mike German both for his contribution to assembly politics since May 1999, and for his hard work as a cabinet colleague during the course of the Labour/Lib Dem coalition."
Plaid Cymru leader Ieuan Wyn Jones also paid tribute to the outgoing leader before congratulating his replacement on her success.
"He has been leader of his party since the first assembly election in 1999 and in this time has had a large impact on his party and Welsh politics," he said.
"I would also like to congratulate Kirsty Williams on her victory. She is an experienced AM and I'm sure that she will be a worthy opponent in the assembly chamber."
Welsh Conservative leader Nick Bourne AM added: "I congratulate Kirsty on her success and wish her well in her new role.
"This is an important moment for the National Assembly and for Wales in general, representing as it does a significant breakthrough in the representation of women in frontline Welsh politics."
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