The Reds produced a display of clinical efficiency on the south coast, stemming the threats of the hosts and striking back through Luis Suarez, substitute Raheem Sterling and Steven Gerrard's late penalty.
Rodgers' team leapt two places into second position in the standings with 10 fixtures remaining.
"It's my job," the manager reflected after the final whistle. "My ambition is for the club - I want the club to succeed.
"We've made many changes on and off the field in the time I've been here, but what we have created is a one-club mentality. Now when anyone talks about Liverpool, it's about the football, and that's very pleasing.
"We still have a lot of work to do in terms of where I want to take the team. It's great to have a group of players that are very honest and want to go with you on the journey. You can see by their work-rate, intensity and quality that we're improving all the time.
"We'll stay calm. It's good for the supporters to dream. It's great for them because they've seen the evolution of this team over the last 18 months - and we've been constantly for over a year progressing and performing at a good level.
"We stay calm, we don't get too carried away and we always just focus on the next game. After Swansea we knew this was going to be a really difficult game here today.
"They've scored in all but two games here, have some wonderful players and the manager is excellent. For us to come and play as we did today and find the solution to nullify that threat was impressive."
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