Monk made 10 changes for the third-round trip to the KC Stadium, giving some much-needed game time to a number of squad men.
But some wasteful finishing saw the visitors draw an unlikely blank against their Sky Bet Championship opponents, who sealed victory with David Meyler's close-range effort in the first half.
Now the likes of Leon Britton, Matt Grimes, Kyle Bartley and Eder have lost the chance of another run-out in the fourth round and must face a battle to force their way into Monk's Barclays Premier League side.
"All these players are first-team players and they're all ready to compete now," said manager Monk.
"But only 11 can start so not everyone can be involved
That's part of your life as a professional footballer.
"It's your job to be professional every single day, work as hard as you can every single day.
"That's the standard I set, whether you're a starter or someone who hasn't been playing
You can be called on at any moment and your job is to be professional for the club that pays your wages.
"I've no problem with this group doing that."
Britton, who captained the side in East Yorkshire, was disappointed to bow out of the competition Swansea won in 2013 but has led calls to make amends against Southampton at the weekend.
"It's very frustrating as we wanted to do well in the cup
We know how great this competition is and we had ambitions of making a real go of it," said the 33-year-old.
"But there's no time to dwell on this
Our focus quickly moves to Saturday's game and we must mentally prepare for it as well as tactically."
Hull defender Curtis Davies took the man-of-the-match award for a strong showing at centre-back, the highlight being a last-ditch goal-line block to deny Gylfi Sigurdsson.
Davies was initially named among the substitutes but was promoted due to Alex Bruce's groin niggle.
"I only found out three or four minutes before we went back into the dressing room after the warm-up," said Davies.
"We were messing around playing keep-ball when Harry Maguire came over to tell me I better do a full warm-up
I thought he was over-exaggerating but then I saw Alex going down the tunnel
I managed a couple of stretches and then I was in, there was no time to think about it.
"We're delighted to come through with a win
Getting through is the only objective in cup football and now we have to rest up because we have another big home game (against Blackburn) on Saturday."
Source : PA
Source: PA