Chelsea found themselves trailing after just 14 minutes when Alexandre Lacazette emphatically found the net from close range.
Laurent Koscielny doubled the lead just before the break with a fortuitous goal as his attempt to meet the ball with a header instead went in via his shoulder.
Chelsea did go close to making things interesting, as Alonso hit the post with a header and then forced Bernd Leno into a late save, but Arsenal ultimately ran out comfortable winners.
Maurizio Sarri was scathing of his team after the game, saying they are difficult to motivate, and Alonso felt the outcome had as much to do with Chelsea's performance as it did Arsenal's.
"We made silly mistakes," he told the club's website. "We provoked them to gain confidence and come to press us because we were making mistakes.
"In the second half we tried, but it was too late and we couldn't score. When you lose, you learn. We have to analyse the game and improve quickly, because there is no time to think. We have to perform again in four days [against Tottenham in the Carabao Cup semi-final second leg on Thursday].
"It was a good opportunity, we wasted it and now we have to work even harder to stay in a Champions League positions and win games. It has to motivate us."
Chelsea are still fourth in the Premier LeAague after Saturday's defeat and are now three points ahead of Arsenal and Manchester United.