Laird, who has been with the Reds since the age of 10, is one of seven Academy graduates to make their first-team debut this season.
The defender made his senior bow in the Europa League tie against Astana and also featured against AZ Alkmaar at Old Trafford in December.
The deal will keep Ethan at the club until June 2023, with the option to extend for a further year.
Woolston, who joined United in 2018, signs a one-year extension as he looks to continue his development with the club.
The North Shields-born goalkeeper has previously represented England at Under-17 and Under-18 level.