Does Jesus say we are saved by faith alone?

Yes, Jesus says we are saved by faith alone. Here are two examples of Jesus making the proclamation that salvation is by faith, in Him, alone.

“Then He said to her, ‘Your sins are forgiven’. And those who sat at the table with Him began to say to themselves ‘Who is this who even forgives sins?’ Then He said to the woman, ‘Your faith has saved you. Go in peace’” (Luke 7:48-50).

“For God so loved the world that he gave his only begotten Son, that whoever believes in Him should not perish but have everlasting life” (John 3:16).

The word faith in the first verse is (Greek) pistis, and the word believes in the second verse is (Greek) pisteuo. Take notice that they have the same root-pist. John uses believes in his gospel just as Paul uses the word faith in his epistles. The way John uses it, basically gives it the same meaning as the word faith seen elsewhere in Scripture. Therefore, Jesus, using both words, is saying that we are saved by faith, in Him, alone.