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Adam Lallana and Jordan Henderson scored to secure a first Premier League win for the team since late August.
Prior to victory over West Brom, last year's Premier League runners-up had won just two of six league games.
"It was satisfying. We had to show character and the resilience of the players was fantastic," said Rodgers.
The Northern Irishman praised the Anfield crowd for giving his team "a lift" after the visitors equalised through a controversial Saido Berahino penalty.
Rodgers also predicted his team's form would improve, with Daniel Sturridge, Joe Allen and Emre Can expected to return from injury following the upcoming two-week international break.
Liverpool's next match will be against bottom-of-the-table QPR on 19 October.
"Certainly from now, he'll have the course of the international break to get ready, and he'll be fine for QPR," Rodgers said of Sturridge, who has missed seven matches since injuring his thigh while on international duty with England
The last Liverpool match Sturridge featured in was the 3-0 victory over Tottenham in August - their last league win before victory over the Baggies.
"We showed great potential against Tottenham and once we get all the players back and we can rotate the squad a bit better that will help us," added Rodgers.
Before the match the former Swansea manager had admitted that his team were unrecognisable from last season's title challengers.
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It was far from a polished performance against West Brom, but the 41-year-old said the three points were all that mattered.
"As we work more together the quality of our game will improve and the performance will improve but you have to find a way to win and we did that," he said.
"That was against a team coming in with good confidence having won 4-0 last week (against Burnley).
"They keep clean sheets, they are difficult to beat and we found a way to break through that resistance and win. It was vital for us before the international break."
Rodgers also questioned Michael Oliver's decision to award the visitors a penalty when Dejan Lovren appeared to foul Berahino on the edge of the box.
"It was never a penalty," said Rodgers. "Thankfully we went on and maybe had a penalty ourselves but luckily for Michael we scored."
Liverpool manager Brendan Rodgers described his team's 2-1 victory against West Brom as a "big win" after a stuttering start to the season.