Palmer was sent off for two bookable offences during Chelsea's 2-0 loss to their London rivals on Saturday as Spurs won a Premier League match at Stamford Bridge for just the second time in 36 years.
He was initially booked for barging into Mateus Fernandes before, with the Blues 1-0 down, Palmer was sent off 10 minutes from full time following an ugly stamp on Conor Gallagher.
"We spoke, and for sure he (Palmer) knows the mistakes he made and things that you can avoid," Alonso told reporters on Tuesday on the eve of Chelsea's League Cup fourth-round tie away to top-flight rivals Liverpool.
"I hope that this is not going to be repeated and he's aware of that, he's mature and he knows what he's doing."
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Palmer started the season in blistering form but has not scored for five games and will miss the trip to Anfield due to suspension.
Alonso, though, said the 24-year-old remains in high spirits.
The Spaniard added: "He's in a good space, he connects very well socially with his teammates.
"They like him, they want Cole smiling, they want Cole having fun, enjoying his game, because when we see that, we have a better day."
Alonso, meanwhile, said forward Joao Pedro will be sidelined for "a few weeks" due to a thigh injury.
The striker, who has scored five goals in the Premier League this season, did not feature against Spurs.
Alonso said the Brazilian felt something in training and had "a lot of fluid" in his thigh.
"He has an injury on his thigh," Alonso explained. "He's going to be a few weeks out. Let's see how many. We don't want to predict on that, but not ready for the next few games.
"He's a key player. We will miss him. But we have to be ready to deal with the injury and be competitive and win games with the players we have available, starting tomorrow.
"This is the situation. We're ready to fight for it and show a good level."