Moyes was appointed as Sir Alex Ferguson's successor last summer, but struggled to continue his Scottish compatriot's success as United lurched from one disaster to another under his guidance.
A string of unwanted records were racked up under Moyes, with Newcastle's 1-0 victory at Old Trafford back in December giving the Magpies their first win at the home of United since 1972.
Their 2-1 reverse against West Bromwich Albion in September was the Baggies' first away win at United since 1978 and Moyes failed to register a victory against Chelsea, Liverpool, Manchester City, Everton or Spurs before he was sacked in April.
Ryan Giggs took over as the interim boss for the final few weeks of the season but they eventually finished in their lowest-ever Premier League position of seventh - a whopping 22 points behind eventual champions, and fierce local rivals Manchester City.
However, Rodgers believes that Moyes was not given enough time to get it right at Old Trafford, after signing a six-year contract last year.
He told The Guardian: “I think there were a lot of things at Manchester United that were not seen by many. He had to go in and where I felt for him is that he thought he was getting time.
"All of a sudden when it’s not as hunky-dory as people think it is, you don’t get that opportunity. He signed a long-term deal and thought he had that longevity to do his work."
The Northern Irishman also believes that the club will approach the current season with a "totally different mind-set" following the arrival of Louis van Gaal.
"I think for them it will be a totally different mind-set," added the Liverpool boss.“Domestically for the last 20 years they’ve been champions 13 times. So they have already been in place.
"When I came to Liverpool we were eighth at the time and the club was nearly going out of business, so it’s totally different. The players there will want to do better for themselves.
"They will be disappointed with where they finished last year, of course. With the quality they have, they will want to push on. "
Van Gaal's side beat the Reds 3-1 in the final of the International Champions Cup on Monday in Miami's Sun Life Stadium to conclude the pre-season friendly tournament.