Goals in either half from Victor Moses and Daniel Sturridge proved enough to settle the tie but the Cherries gave a good account of themselves as they outplayed their Barclays Premier League opponents in large spells.
Rodgers, who named a strong side for the tie including the likes of Steven Gerrard and Luis Suarez, believes the kind of football Bournemouth play is a testament to 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," said 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."
It would have come as a surprise to many supporters that Rodgers elected to field such a strong side, which showed just three changes from the 2-2 draw at Aston Villa last week.
Liverpool face Merseyside rivals Everton on Tuesday night as they look to keep up the pace of the top three sides but Rodgers wants to go all the way in the FA Cup and was happy to send out his big hitters.
"We are a very thin squad and we have a number of injuries," he added.
"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."
Sturridge, who scored his third goal after returning from an ankle injury, praised the work ethic of the Liverpool players to ensure they came out on top.
"We played some good football," he told ITV Sport.
"In the first half we didn't really defend as well as we could have, put our stamp on the game, but we did work hard, in the end we worked hard and our quality insured we won the game."
Howe was left bemoaning the lack of fire-power in his side, which was highlighted as Liverpool scored both of their goals during periods of sustained Bournemouth pressure.
"I don't think it [a positive result] was far away," he said.
"Everyone says goals change games and they do because with their first real attack they have scored and showed the cutting edge that we didn't unfortunately."
Liverpool had been knocked out at the hands of Oldham last season, who had a much more physical than Bournemouth's approach this time around.
Howe insists he will always try and play football, whoever the opposition may be.
"Our players are not built that way and we don't recruit that way," he said when asked if a more physical game would have offered more of a challenge to Liverpool.
"We are trying to build a club that is club here that is built of possession very much like Liverpool, we very much want to go down that route.
"This was a chance for us today to go toe-to-toe with a really good football team and see if we could beat them footballing-wise.
"On another day with better decision making in and around the box we could have led at half-time."