Chelsea dominated from the opening whistle, with the forward line of Estevao, Cole Palmer and Morgan Rogers – behind Welbeck – playing some exhilarating football.
But fine defending and spectacular goalkeeping from Josh Keeley kept the Hatters competitive for most of the match.
The first half saw Keeley deny Palmer and Reece James, and Rogers hit the bar with a volley. But the deadlock was eventually broken when Welbeck, playing for his seventh club at the age of 35, leapt to head home a fine cross from Malo Gusto.
Palmer also hit the bar in the second half, but although there was little to do for Mike Penders – keeping the spot between the sticks warm for potential new signing Emiliano Martinez – Luton remained in the game.
The tie was put beyond doubt when Estevao turned outside Joe Johnson and shot towards the far post where Hakeem Odoffin, attempting to clear, kicked into his own net.
The Blues will face Leeds United in the third round in the week commencing September 7.
