The Dutchman was pegged back at one set each but Jenkins, seeded 16, was unable to convert a 2-0 lead in the vital fourth set as Van Gerwen eased through.
Van Gerwen said: "I am up to the quarter-finals and I feel confident. There is still something in the tank."
Robert Thornton is also in the last eight after beating Benito van de Pas.
Ninth seed Thornton of Scotland won 4-0 and will now face Van Gerwen at Alexandra Palace.
Dutch five-times world champion Raymond van Barneveld scraped through to the last 16 with a 4-3 win over Derby's Jamie Caven, the 19th seed.
BDO world champion Stephen Bunting also reached round three by upsetting sixth seed and fellow Englishman James Wade 4-1.
England's Michael Smith came back from 2-0 down with 10 maximums to beat Northern Ireland's Brendan Dolan 4-2 in the second round.
Dutchman Vincent van der Voort completed a 4-0 whitewash over unseeded German Max Hopp to reach the third round.
Michael van Gerwen stayed on course for a successful defence of his PDC World Championship with a 4-1 win over Terry Jenkins to reach the quarter-finals.