Rogers needed just seven minutes to net his first goal in a Chelsea shirt, pouncing quickest after Alex Remiro pushed out Joao Pedro’s initial header.
The £117m club-record arrival from Aston Villa looked sharp early on, linking neatly with Cole Palmer and Pedro in Chelsea’s front three.
Within 10 minutes of his debut, the home crowd were already chanting his name.
It was no surprise he faded as the game wore on—he had trained for less than a week after returning from holiday—before coming off after 62 minutes.
Aside from Rogers, Joao Pedro has been the standout of Chelsea’s pre-season.
Here he restored Chelsea’s lead with a 47th-minute header, then wrapped up the win with a composed side-footed finish 30 minutes later, after Sociedad had levelled just before the break through Jon Mikel Aramburu’s volley.
Chelsea lost their grip late in the first half and looked exposed, but they were far more assured after the interval.
Pedro now has eight goals in seven pre-season games and led the line with Rogers and Palmer operating just behind him.
Alonso offered a glimpse of the 3-4-2-1 he might use at Fulham in Chelsea’s Premier League opener, as he took charge at Stamford Bridge for the first time.
Robert Sanchez started in goal, with a back three of Wesley Fofana, new signing Maxence Lacroix and Levi Colwill from right to left.
Pedro Neto filled the right wing-back role and Marco Palestra the left, while captain Reece James and Moises Caicedo anchored midfield behind the front trio.
Substitute wing-back Pep Chavarria made his debut, though several notable players were missing.
Alonso maintains he won’t be tied to a back three or four, but this was the same setup used when Chelsea beat AC Milan 3-0 a week ago.