United put together a nine-game unbeaten run in the league, which included six wins, after the coronavirus lockdown to pip Chelsea to third place at the end of last season.
But they finished 33 points behind Liverpool, who claimed their first league crown in 30 years, and 15 points adrift of City.
Rashford said chasing down their two rivals gave United added motivation.
he told the Press Association.
I think that's natural for anyone who plays for the club or anyone who's from Manchester,
It's obviously a big step," he said. "The lads worked hard and, for me, it's what we deserved as a team. We've had some tough times (last) season and some times where we've been unstoppable.
We have to use all that as motivation to keep improving as a team.
United have only finished higher than they did last season once since they won their 20th title in 2012/13, when they were runners-up under Jose Mourinho two years ago.
Rashford is convinced they now have something to build on.
The England striker missed a big chunk of the last campaign with a back injury but said he was disappointed with his tally of 22 goals for United.
said the 22-year-old Coca-Cola ambassador.
I missed four or five months and in my mind, I could have done much more... but for whatever reason, it didn't happen,
"I never lose touch with reality and I know next season, I have to come back better than I was last season.