Goals from Luke Thomas and Caglar Soyuncu earned the Foxes three crucial points in the race for the top four.
Leicester moved back into to third with the victory, up to 66 points after 36 games, sitting ahead of fourth-placed Chelsea on 64 points from 35 games.
West Ham and Liverpool are on 58 and 57 points from 35 and 34 games respectively, with Leicester's win creating an improbable deficit to overcome.
The Foxes crumbled at this stage in their Champions League bid last season so the magnitude of a rare win at Old Trafford was not lost on Rodgers
"It’s an extremely important win for us," Rodgers said. "It’s huge.
"We’re at 66 points now, which is what two of the sides that finished in there last season got.