Hogan's finish 15 minutes from time sealed the win for Stoke, who went ahead early in the second half courtesy of a bundled effort by the impressive William Forrester.
Having said that, Forrester and Scott Wara performed well at the back for the away side and Stoke were deserved winners on the balance of the 90.
The game got off to a scrappy start and it took ten minutes for the first chance to fall to Fulham's Carvalho.
The diminutive playmaker was played in by Taylor-Crossdale, but he shot tamely and Stoke goalkeeper Blondy Nna Noukeu was able to save comfortably.
Stoke gradually began to grow into the game and on 17 minutes they had their first chance when Mohamed Sankoh was able to break following a Fulham corner.
Sankoh created space to shoot on the edge of the box effectively, but his effort was too high.
The half continued to be scrappy throughout, with Jayden Harris making a particularly bad challenge on Stoke defender Wara that earned the Fulham man a yellow card.
The best chance of the half fell to Sylvester Jasper on the half hour mark, when he latched onto a through ball, skipped inside Tre Pemberton and shot towards the bottom corner of the Stoke goal.
However, Nna Noukeu was able to get a hand on the effort and turn it away for a corner.
Hogan then had a decent opportunity for Stoke following an excellent cross from Ryan Corrigan, but his header was tame and straight at Fulham keeper Taye Ashby-Hammond.
Just before half time Fulham had a couple of free kicks in decent areas, but neither Carvalho nor Ryan De Havilland were able to test Nna Noukeu.
The first chance of the second half fell to Kalvin Lumbombo-Kalala after a good through ball by Fulham striker Timmy Abraham, but his shot failed to test Nna Noukeu.
Minutes later and players from either side were booked following a large fracas, which started when Cody Drameh upended Corrigan.
Corrigan and Drameh were both booked, while Abraham and Pemberton also received a caution for their involvement.
Stoke finally broke the deadlock on 58 minutes, when Forrester bundled the ball in following a series of failed clearances from Fulham.
The ball initially came in from a dangerous Pemberton cross and Forrester was on hand to finish after Hogan had his initial effort blocked.
With Stoke ahead they were able to sit back more comfortably than they had previously and it became increasingly difficult for the home side to create chances.
Both Forrester and Wara made a number of good clearances, while Sankoh continued to offer a threat on the break for the Potters.
And it was on the break that Stoke managed to make it 2-0, with Hogan running onto a through ball and finishing with the outside of his foot.
Hogan had been largely anonymous throughout the match, but it was a moment of quality from the former Brentford man that allowed him to double Stoke's lead with just 15 minutes to play.
Fulham nearly had a response within minutes through Carvalho, who forced Nna Noukeu into a save from range.
Luckily for the Stoke keeper, he parried the shot just out of reach of Abraham, but it wasn't the most convincing piece of goalkeeping from Nna Noukeu.
Five minutes from the end and another mistake by Nna Noukeu presented Taylor-Crossdale with the ball inside the Stoke area, but the keeper was just about able to recover and block the midfielder's shot at the last second.
Carvalho went close once again late on, but a Stoke defender was on hand to block and preserve their clean sheet.
Fulham: Ashby-Hammond, Drameh, Mundle-Smith, Harris, Asare, De Havilland, Lumboboa-Kalala, Taylor-Crossdale, Abraham, Carvalho, Jasper
Substitutes: Ribera Palacios, Benjamin, Lanquedoc, Olakigbe
Stoke City: Nna Noukeu, Pemberton, Corrigan, Wara, Forrester, Tymon, Dunwoody, Sorensen, Hogan, Duffy, Sankoh
Substitutes: Macari, Kanuric, Porter, Sankoh, Jennings
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