Crawley, who impressed Leeds in the third round, leveled thanks to Tom Nichols’ close-range finish but King’s placed effort saw the hosts through.
After collecting Josh King’s backheel Wilshere, starting for the first time in his second spell at the club, curled in from 20 yards.
On Wednesday, 10 February, Bournemouth will go to Burnley in the fifth round.
AFC Bournemouth – Crawley Town 2-1
Goals: Wilshere (24′), Nichols (59′), King (65′)