"I promise I'll fight for my life and for the people in this city," said the Northern Irishman.
Gerrard keen to meet Rodgers
"We might not be ready for the title now but the process begins today."
Swansea are understood to have agreed a £7m compensation package for losing the services of Rodgers, coach Colin Pascoe, assistant performance analyst Chris Davies and performance consultant Glen Driscoll. Rodgers is believed to have signed a three-year deal.
Liverpool have won the title 18 times in their history but the last time they were crowned champions was 1989-90.
"This is long-term and that's the appeal to me," added former Watford and Reading boss Rodgers.
"The history of the club and the frustrations was also an attraction. It has been 20-odd years since they won the title.
"This is a club that, if you have success, you are here for years.
"That's the aim arriving here. Of course that is about results and the progress of the team.
"I am very proud. I feel I have been blessed with the opportunity to manage the club."
Who is Brendan Rodgers?
Born 26 January in Carnlough, Northern Ireland
Begins career as a defender with Ballymena United
Joins Reading as a teenager before being forced to retire through injury. Remains as a coach
Jose Mourinho appoints Rodgers as Chelsea youth team manager
The Blues promote him to reserve team manager
Becomes Watford boss
Returns to Reading to replace Steve Coppell
Leaves by mutual consent after a bad start
Returns to management with Swansea
Guides the Swans to promotion via play-offs
Leads Swansea to 11th in their first Premier League campaign
The club have also scrapped plans to appoint a director of football - at Rodgers's request. Instead the club will create a "committee" which will make decisions on the running of the football side, including transfers.
"That [a director of football] was something I made clear I couldn't work with," added Rodgers. "What you need is an outstanding team. We will form a technical board that will have four or five people that will decide the way forward."
Liverpool fans have given a mixed reception to the appointment of Rodgers but the new man in charge at Anfield has vowed to win over the doubters.
"Hopefully over time I will earn their respect," he said. "It's a real special club. I'm really looking forward to moving here to the city."
On how he would handle Liverpool's big-name players, he said: "It is about people. It isn't about stars. I treat them as human beings.
"It is about respect and I have carried that from coaching kids at five years old to some of the biggest stars in the world. I was never the biggest player so I have had to earn my respect on the coaching field and as a human being.
"I am really looking forward to working with some of the greats of this football club. For me it will take a bit of time to introduce how I want to play and the philosophy I want to introduce.
"I believe there is still a bit of work to do in maybe getting three or four key players in, to allow us play that way. I had a great chat with Jamie Carragher on the way down and he's really excited about playing this way, as was Steven Gerrard when I spoke to him.
"We talked finances. The reality is that I'm not going to have the money that was spent in the last year or so. But, at the same time, there's going to be money there to make a difference."
Rodgers did not comment on reports that an agreement is in place to prevent him from signing players from his former club Swansea for 12 months.
But Swans chairman Huw Jenkins told talkSPORT: "We have got some protection on Brendan coming back for our players in the initial period, which I think is the right thing."
"By the time the gospel according to Brendan Rodgers had drawn to a close he was preaching to the converted."
The future of Steve Clarke, who was assistant to Dalglish, will become clearer in the next few days.
Liverpool's principal owner John W Henry tried to ease the pressure on Rodgers, and said: "We do not expect miracles overnight, nor should anyone else.
"In Brendan Rodgers we are recruiting an exciting, young and talented manager. He is a forward-thinking coach at the forefront of a new generation of managers. He will bring a style of relentless attacking football. He is someone we think our players will respect and our supporters will embrace Brendan's style of football.
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