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Caretaker manager Michael Appleton was in the dug-out at the Liberty Stadium to see his side end a run of six Premier League games without victory.
The visitors went 2-0 up either side of half-time through a Federico Fernandez own goal and a Shinji Okazaki effort, with Alfie Mawson replying for Swansea.
It was Leicester’s first away win since April and took them out of the relegation zone.
Shakespeare was dismissed just four months after succeeding the sacked Claudio Ranieri, architect of Leicester’s fairytale title triumph in 2016.
Appleton made one change to the team that had drawn 1-1 at home to West Bromwich Albion in Shakespeare’s last game, with Okazaki replacing Kelechi Iheanacho.
The visitors made a storming start and Swansea goalkeeper Lukasz Fabianski was obliged to save from Okazaki, Marc Albrighton, Jamie Vardy and Wilfried Ndidi in the early stages.
Leicester duly went ahead in the 25th minute when Fernandez headed Riyad Mahrez’s cross into his own net.
Okazaki added a second early in the second half, tapping in after Albrighton’s cross was steered into his path by Mahrez.
Mawson reduced the arrears in the 56th minute when he hooked in from close range following a Renato Sanches corner.
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