Johnson became Peterborough's fourth manager of the season earlier this week but he was unable to prevent Nigel Pearson's visitors from picking up all three points at London Road.
Steve Howard opened the scoring before the break for Leicester before winger Tommy Rowe briefly levelled matters for Posh.
Andy King soon had the Foxes back ahead and they held on to keep pace with the play-off pack.
Both sides were promoted from League One last May but their seasons have gone in different directions since, with the Foxes in the hunt for a second straight promotion and Posh going down with a whimper.
United were relegated on Monday and have lost 15 of their past 20 games in all competitions but they started brightly, with Dominic Green scuffing an early shot straight at Leicester goalkeeper Chris Weale.
Foxes left-back Bruno Berner then crossed for Paul Gallagher to fire over from the edge of the box before Posh winger Sergio Torres danced through the defence but failed to find the finish as Weale made an easy save.
Craig Morgan then had a header cleared off the line by Berner as Posh desperately looked for their first goal of the Johnson era - but it was Leicester who broke the deadlock in the 40th minute.
Green lost possession in his own half and Leicester launched a breakaway through Richie Wellens.
Howard received the ball wide on the right and after beating Ryan Bennett he drove a low shot across goalkeeper Joe Lewis and in off the far post.
Johnson introduced Ben Wright at the start of the second half as he went with two central strikers and the hosts were rewarded for their positive outlook when Rowe equalised in the 63rd minute.
Josh Simpson played a ball into Wright on the corner of the box and he teed up the onrushing Rowe to strike powerfully beyond Weale for a fine goal.
But the relegated side were back on level terms for just three minutes before King struck.
Gallagher delivered a teasing cross from the left and after substitute James Vaughan sent the ball back across goal, King was on hand to head in an easy finish.
Craig Mackail-Smith had a chance to scramble in a late equaliser but Weale made a brave save to keep Pearson's side firmly in the picture at the top of the table.