Shakespeare's fourth successive victory in all competitions since he took over from Italian Claudio Ranieri last month left Leicester in 14th position on 30 points from 28 games, six above the relegation zone.
The visitors produced a superb first half as early goals by Riyad Mahrez and centre back Robert Huth gave them a 2-0 lead and Jamie Vardy made it 3-1 after Manuel Lanzini had pulled one back with a brilliant free kick.
The Foxes were on the back foot after Andrew Ayew headed in West Ham's second and goalkeeper Kasper Schmeichel producing two stunning saves to deny Andy Carroll while Ayew blasted over the bar with the goal at his mercy.
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(Writing by Zoran Milosavljevic, editing by Ed Osmond)
