Crawley were briefly level in the second half thanks to Tom Nichols' equaliser but their preparation for this tie was unsettled to say the least.
This was their first outing since a magnificent defeat of Premier League side Leeds, due to a Covid-19 outbreak at the Sky Bet League Two club.
They are still suffering the effects of that and, despite this fixture being moved from its scheduled slot last weekend, Crawley were still missing five players due to coronavirus.
Bournemouth had not won for three games and have slipped in their quest for promotion to the Premier League.
Wilshere was making his second full debut for the Cherries after rejoining Jason Tindall's side following his release from West Ham in October.
Philip Billing then struck a post with a header from David Brooks' corner, whilst Crawley's sole opportunity was when Ashley Nadesan saw his powerful effort saved by Asmir Begovic, with Nichols unable to reach the rebound.
Bournemouth scored in the 24th minute when an incisive pass from Danjuma and a clever touch from King allowed Wilshere to sweep home a from the edge of the area, his first goal in 18 months.
They might have had more, with David Brooks seeing his goal-bound shot blocked on the line but John Yems' side made it to half-time only one goal behind.
Four minutes into the second half, Wilshere almost added his and Bournemouth's second as he hooked a shot from close range which Morris saved superbly.
Crawley's best chance came when Begovic dithered on his goal line and Nadesan got a foot to his attempted pass. Nichols seized on it but Begovic recovered quickly to block his shot.
Nichols then had his penalty appeals waved away as he tumbled under Lloyd Kelly's challenge, but a minute later he had equalised.
A well-worked move culminated with Josh Wright heading the ball to Nichols and he finished with a side-footed volley from close range.
The League Two side were only level for six minutes however, as King crisply put Bournemouth back ahead after Danjuma and Brooks had opened up the defence.
It was the Norway international's third goal of the season after scoring twice against Oldham in the third round.
Bournemouth missed numerous chances to make the end of the game more comfortable, perhaps the best of which was when Steve Cook headed wide.
But they had done enough to secure a fifth-round trip to Burnley.