The hosts conceded in second-half injury-time to force an extra 30 minutes, and then had their defence breached again with seconds remaining on the clock before the game was heading to penalties.
For much of the game it looked like there was going to be another upset in the famous old trophy, as the National League side defended valiantly against the fifth placed Premier League side.
Unsurprisingly, David Moyes' side dominated possession throughout as they looked to break down a stubborn Kidderminster defence.
However, despite having most of the ball, it would be the Champions League chasing team who would concede the first goal of the match.
A set-piece from Omari Sterling-James on the left side of the pitch brought a lot of disarray in the West Ham penalty area, and despite his best efforts in attempting to catch the ball, Alphonse Areola produced a howler by dropping it right to the feet of Alex Penny who netted into an empty net from seven yards out.
On co-commentary for talkSPORT, Irons legend Alvin Martin was not happy with what he had seen, but said the hosts had an element of fortune.
"I expected the delivery to be good, not that good, they got lucky," he said.
"The ball bounces in their favour, you can't do anything about the finish, he did really well.
As the match progressed, the tempo remained the same.
West Ham had most of the ball looking to find a way to break through the Kidderminster rear-guard whilst the hosts looked to make the most of counter-attacks and set pieces.
The game plan was working perfectly for Russell Penn and even when Moyes brought on his star players such as Declan Rice and Bowen, the hosts remained firm.
However, in the dying seconds of the match hearts were broken as the Manchester United transfer target produced a goal to save the Hammers' blushes.
Skipping inside into the penalty area, the vice-captain did brilliantly to elude challenges from the Kidderminster defence before smashing the ball into the roof of the net.
Understandably delighted with the goal, Rice felt the need to produce a rather unique celebration by recreating a Michael Jackson dance.
Martin, alongside every West Ham fan were delighted they avoided embarrassment, and said that the goal might add a bit more to his transfer value.
"What a goal," the former centre-back said. "It might have been a substandard performance by West Ham, but that was a moment of class from Declan Rice.
"He took the game by scruff of the neck, a quality goal by a quality player when it was needed most.
"You can't help but feel for the Kidderminster supporters, who have sunk. They were seconds away from round five.
"I thought there would be a chance, but that wasn't a chance, it was made by Rice. There was nothing on when he picked the ball up.
"Well done to David Moyes for putting him on the pitch. He shouldn't have had to, but that's testament to how well Kiddermister played.
"West Ham have had to rely on an individual. West Ham were in desperate trouble, but he saved them.
"It will add a few more millions to his price tag, that's for sure."
With an extra 30 minutes to come, the hosts had to summon even more energy to try and still cause an upset.
Whilst they didn't threaten much in the period, every player stood up to the endless attacks from the Irons, and looked like they had done enough to force a penalty shoot out.
However, with seconds to go before the game would go there, the Liverpool transfer target ensured West Ham's name would be in the hat for the fifth round of the competition.
Andriy Yarmolenko, in typical fashion, cut the ball back onto his stronger left foot and unleashed an effort that ricocheted its way through the penalty area before the ball broke to Aaron Creswell, who crossed the ball into the six-yard box for Bowen to have the simple task of tapping in from close range.