The Reds host Everton in the Merseyside derby in the Premier League on Tuesday night but Rodgers still fielded stars like Steven Gerrard, Luis Suarez and Daniel Sturridge in Saturday's fourth-round tie at the Goldsands Stadium.
In total, Rodgers made just three changes to the side that drew 2-2 at home to Aston Villa last weekend.
Liverpool, fourth in the table, lie six points behind third-placed Chelsea, and only one point above their city rivals - so there is little capacity for slip-ups.
But despite having several players in the treatment room, Rodgers is desperate to take the club back to Wembley this season.
"We are a very thin squad and we have a number of injuries," said the Reds boss.
"The tradition in the FA Cup for Liverpool is huge and we want to pay respect to the competition because we want to get to the final and win it if we can."
Liverpool made it through to Sunday's fifth-round draw courtesy of goals from Victor Moses and Daniel Sturridge in either half.
However, the Cherries gave a good account of themselves as they outplayed their opponents in large spells.
Rodgers believes the kind of football Bournemouth play is a testament to their manager Eddie Howe and thinks the 36-year-old has a big future in the game.
"When you have a team that want to play football, they are the most courageous players on the field," added Rodgers.
"It is easy to coach a team to sit back, not want the ball and defend and then just wait for a hump up the pitch.
"The courage that the Bournemouth players showed today was fantastic - they have got better, I have watched them as the season has gone on.
"It is coaches like Eddie Howe who will take the game forward in this country because they believe in a way of working and they have a philosophy.
"Bournemouth are a very good side and I give credit to them first of all. Eddie is similar to myself in terms of the philosophy of football.
"It is not easy when you're a young manager to stick your head out to get your teams to play and pass the ball but he certainly does that."