After starting the new Xabi Alonso era with a 3-2 Premier League victory at Fulham, Chelsea made several changes for the visit of the League One side. Welbeck was given his opportunity and the experienced striker made it count.
The 35-year-old finally broke Luton's resistance early in the second half when he rose to meet Malo Gusto's cross and headed the ball into the net.
Welbeck believes his experience can provide an important influence in a Chelsea squad containing plenty of young talent.
"I bring the average age a little bit higher!" Welbeck said.
He added: "The experience of playing games, being around loads of different dressing rooms and knowing what is a good dressing room and a bad one. Knowing what it takes to achieve something special."
Chelsea doubled their advantage when Luton defender Hakeem Odoffin turned another dangerous delivery from Estevao into his own net.
The victory sends Chelsea into the last 32, where they will face Leeds at Stamford Bridge.
Mike Penders was handed his first start in goal after Robert Sanchez endured a difficult outing against Fulham.
Chelsea are also reportedly closing in on a deal for Argentina goalkeeper Emi Martinez, who could add further experience to Alonso's squad.
However, Enzo Fernandez was absent amid reports linking the midfielder with a move to Manchester City.
Alonso said he wants Chelsea to have the right balance between youth and experience as he builds his squad.
"We want to have a complete squad in all aspects. In quality, personality, young and experienced players - so the young players get help, that they are still players to develop," the Chelsea manager said.
"I'm happy with the energy and attitude of the players."
Fulham also progressed to the third round after beating AFC Wimbledon 3-0.
Emile Smith Rowe and Rodrigo Muniz scored in the first half before Gonzalo Garcia completed the win from the bench, scoring for the second time in as many matches since joining Fulham from Real Madrid.
