In a thrilling, end-to-end encounter at Peninsula Stadium, City were two goals down and then 3-2 behind late on, showing their powers of recovery to take the game to penalties, where Jakub Stolarczyk was the hero, saving from Sam Fielding in sudden death to send City through.
Two changes were made to the team that narrowly lost to Chelsea on Friday, with Adam Leathers and Demarai Gray making way for Vontae Daley Campbell and Ali Reghba, who both returned from suspension, as the Foxes reverted to a familiar 4-2-3-1 formation.
City enjoyed large spells of possession in the opening exchanges, though Salford's Dan Hawkins provided the game's first attempt and found Stolarczyk equal to his low effort.
But the Foxes came even closer inside five minutes when Reghba did brilliantly to nutmeg Fielding at the byline and find an angle to get a strike away, which was parried by Vaclav Hladky.
Callum Wright's follow up effort was also stopped by the Salford 'keeper as City missed a golden opportunity to take the lead early on.
Stolarczyk was then called into action again to keep out Bruno Andrade's low strike, as the home side started to come into the game.
And it would be the hosts who broke the deadlock on 24 minutes. Tylor Golden was brought down in the box and although Stolarczyk got down to save Brandon Thomas-Asante's spot-kick, the midfielder converted the rebound.
The Foxes 'keeper became increasingly busy, dealing with Jordan Turnbull's half-volley, despite the ball bouncing up awkwardly, before holding on to Tom Elliott's rebound.
Thomas-Asante came close to scoring for the second time just after the half hour mark when he got in front of the Leicester defence to meet Andrade's low cross, but couldn't find the target from close range.
City continued to have plenty of the ball in and around the Salford box, but failed to create too many clear cut opportunities, bar Callum Hulme's low attempt, which was saved by Hladky.
And just before the break, the play opened up for Wright to lay off Khanya Leshabela, who dragged his low attempt wide of the post.
Into the second period, Admiral Muskwe got in behind the Salford defence and hit the side netting from a tight angle and City were made to pay for that miss.
When half-time substitute Mani Dieseruvwe raced towards goal, he provided a cool finish into the bottom corner, doubling the hosts' advantage against the run of play.
Just as Salford had got a hold of the game, City gave themselves a lifeline, grabbing a goal back almost immediately. Reghba tried to take the ball around Hladky, who became stranded outside his area and presented Muskwe with the chance to finish. The forward made no mistake, curling the home into the corner.
And Muskwe, who also netted against Chelsea on Friday, scored his second in five minutes to level up the tie. Thanawat Suengchitthawon, a half-time substitute, carried the ball forward and into the area before laying off Muskwe to provide the finish from close range.
The Foxes looked set to take the lead for the first time as Leshabela saw a curling effort deflected just past the post, before Muskwe came close to a hat-trick goal, forcing Hladcky into a diving save after the shifting the ball onto his right foot.
But it was Salford who went straight up the other end and scored the fifth of the contest. Stolarczyk spilled a powerful strike into the six yard box where Dieseruvwe was waiting to tap home for his second of the game.
Leicester showed great character to equalise once more with seven minutes remaining, as the ball fell for Shane Flynn, who lobbed the stranded Salford 'keeper in the closing stages, taking the game to spot-kicks.
After Stolarczyk saved the home side's first from Dieseruvwe, Wright, Callum Hulme and Salford's Loughlan all scored, ahead of Reghba missing from 12 yards to make it 2-2.
Then came a period of eight successful penalties including from City's Muskwe, Suengchitthawon, Kasey McAteer and Leshabela, to move into sudden death.
Stolarczyk then produced his second stop, keeping out Fielding to spark jubilant scenes from the Foxes players and bench.
Major moment…
City had it all to do at 2-0 down and demonstrated their powers of recovery to come from behind twice, but it was the second penalty save from Jakub Stolarczyk that proved to be the match-winning moment and ensured City's progression.
Who impressed?
Admiral Muskwe was a constant threat in the second half and could have had more than two goals to his name, had it not been for the heroics of Hladcky. But the two he did find the back of the net with were both superbly taken, getting his side right back into the game.
Where does it leave us?
The Foxes await their EFL Trophy last 16 opponents and are currently the only Development Squad to have made it through into the next knockout round.
Coming up…
City's Development Squad are back in Premier League 2 Division 1 action on Saturday afternoon, travelling to the south coast to face Southampton Under-23s (2pm kick-off GMT).
The details…
City: Stolarczyk, Daley-Campbell (McAteer 70), O'Connor, Nelson, Flynn; Leshabela, Hulme; Ndukwu (Suengchitthawon 45), Muskwe, Wright (c); Reghba.
Subs not used: Fitzhugh, Leathers, Russ, Odunze.
Goals: Muskwe 64', 69'. Flynn 83'.