On their first UEFA Champions League appearance in five years, Steven Gerrard's late penalty ensured the hosts took maximum points at Anfield having been given a scare by Ludogorets.
Mario Balotelli had opened the scoring with his first goal for the club only for Dani Abalo to level in the final minute of normal time. However, captain Gerrard was Liverpool's hero as he snatched victory late on in a fixture many had envisaged would not pose too many problems.
The win came off the back a surprise 1-0 home defeat to Aston Villa and, as Liverpool prepare to return to league action at Upton Park, Rodgers has urged his charges to build on Tuesday's hard-fought success.
"Against Villa we weren't at our level in the final third and Villa defended well," the manager said on Thursday.
"We moved on from that, got a great result [against Ludogorets] and the perspective after a couple of days makes it a good result.
"We showed the character and the mental qualities that successful teams have to come through and win the game, something we'll look to take into the weekend.
"Sam [Allardyce's] sides are well set up and always make it difficult for you. It'll be a really good test for us...but we'll look to impose our game."
Last season's Premier League runners-up have yet to hit top gear this term after bringing in a host of new signings during the transfer window, including striker Balotelli.
However, West Ham have also strengthened and two of their close-season signings were on the mark in Monday's 2-2 draw at Hull City, with Enner Valencia's stunning strike underlining his threat while Diafra Sakho has been credited with the second.
Allardyce is hopeful of providing Liverpool with a stern test as he seeks to gain a measure of revenge for the corresponding fixture last season, which saw a 2-1 win for the visitors and was packed with controversy.
All three goals required big calls from referee Anthony Taylor - not least Liverpool's two penalties - and Allardyce is keen to make amends this weekend.
"I remember last year we were sadly done by a referring decision," Allardyce explained. "They complained that they had to park too far away, the dressing rooms were too hot, the grass was too long, whatever.
"It was a shame they won on a dubious penalty because we played ever so well. Hopefully we can repeat that minus the penalties.
"We might not get as much possession as we usually do so we'll have to make the most of it, and take the chances we create.
"I'm not daft enough to say we're going to press Liverpool and pin them back in their own half but we're going to try when we can [and] make sure at the same time we're resilient at the back."
Allardyce has no fresh injury concerns following the draw at the KC Stadium while Rodgers, who is once again without Jon Flanagan after the full-back underwent a knee operation, could welcome back midfielder Joe Allen (knee).
