Jerome scored Stoke's second goal as they came from behind to beat Gillingham 3-1 at the Priestfield Stadium, with Jon Walters and Robert Huth also finding the net.
Next month, Stoke will face Valencia in the last 32 of the Europa League, a competition they qualified for by reaching last season's FA Cup final.
Tony Pulis' men are also in the top half of the Barclays Premier League table but Jerome said: "I think it's quite manageable.
"We have had a heavy schedule before Christmas but we have got through that.
"We are sitting in eighth position in the league, which is good, and we are through to the last 32 of the Europa League after no-one gave us a prayer before the competition started.
"We are doing well. We are all sorted at the minute, so hopefully we can progress in this competition as well.
"It's something the manager wants to do and something the player wants to do. Hopefully (we can go one better than last year).
"We'll play our football and look to win games - that's all we can do."
Stoke started slowly at the Priestfield Stadium but responded to Danny Kedwell's 16th-minute strike with three goals in the space of 15 minutes around half-time.
"We don't want to be the ones that slip on a banana skin," Jerome said. "We knew coming here that we could be. We fell one down, but we rallied well, showed great team spirit, got ourselves back in the game and took control in the second half.
"We are in the next round and we want to progress like last year. I'm pleased for the lads and it's a nice win."
Pulis, a former Gillingham manager, relished his return to the Priestfield Stadium - despite being banned from most of the ground by Gills chairman Paul Scully.
Pulis managed the club between 1995 and 1999, guiding them to the brink of the Championship before leaving in acrimonious circumstances, which ended in a court case.
But Scally's insistence that he could never forgive Pulis appeared to leave him in a minority of one as the Stoke manager received a standing ovation from the Gillingham supporters.
"If he (Scally) wants to say stuff, it's a free world. He can say what he wants. You've seen who really counts and that's the supporters," Pulis said.
"I have been a manager a long time and I have never had all four sides of the ground stand up and applaud me when I walk on the pitch, and singing my name at the end.
"It was fantastic. That is what you work for at a football club - the supporters and the people in the area.
"I had a great four years at the club. I can remember my first game here against Wigan in front of about 2,300 people.
"My last game was at Wembley (in the League One play-off final) and we took 38,000 there.
"It has put me in good stead. It made me a better manager. In four years I was able to build a football club that I thought could push on, and did push on.
"That reception will stay with me for a long time, it was very, very special. I didn't realise it was going to happen.
"It has been 12 years since I left the football club. So to be held in that esteem after 12 years - there aren't many managers who get that."
Gillingham put in a spirited effort and continued to cause Stoke problems even after falling 3-1 down, with Swindon target Luke Rooney particularly effective off the bench.
Swindon, who caused the biggest upset of the third round by knocking out Premiership side Wigan, revealed this week they had lodged a "six-figure" bid for Rooney - much to the anger of Gillingham manager Andy Hessenthaler.
"I am not going to get into a war with Swindon but to put that into the press a day before a big FA Cup tie, I am disappointed in their chairman and I know my chairman is livid as well," Hessenthaler said.
"They had a private conversation and to go out on Radio Wiltshire and say they have made a six-figure bid, it is not good conduct. That is how we feel."
Hessenthaler's desire to make Jo Kuffour's loan move from Bristol Rovers a permanent one increased after Kedwell went off with a suspected cracked rib.