Eastman was born in Amherst, Massachusetts to Clarence Willis and Ann Hull (Dey) Eastman. After preparing at Phillips Academy and Williston Academy, he graduated from Amherst College in 1933 and later from the National Academy of Design.

From 1936 to 1941, Eastman worked at the story department of Walt Disney Productions. From 1941 to 1943 he worked at the story department of Warner Bros. Cartoons. From 1945 to 1952 he worked in the story department of United Productions of America. He contributed to the "Private Snafu" World War II training films, wrote for the animation Mr. Magoo, and the Gerald McBoing-Boing series for UPA.

A protegé of Theodor Geisel (Dr. Seuss), Eastman wrote many books for children, in his own distinct style under the Dr. Seuss brand of Random House, many of which were in the Beginner Books series. His titles include:

• Are You My Mother? (ISBN 0-394-80018-4)(Beginner Book #18)
• Go, Dog. Go! (ISBN 0-394-80020-6)(Beginner Book #20)
• The Best Nest (ISBN 0-394-80051-6)(Beginner Book #51)
• Flap Your Wings (ISBN 0-375-80243-6)(Beginner Book #88)
• The Cat in the Hat Beginner Book Dictionary (ISBN 0-394-81009-0)
• What Time Is It? (ISBN 0-375-81362-4)(Bright and Early Board Books)
• Sam and the Firefly (ISBN 0-394-80006-0)(Beginner Book #6)
• Big Dog... Little Dog (ISBN 0-375-82297-6), with additional illustrations by Peter Anthony Eastman(Beginner Book #92)
• Red, Stop! Green, Go! (ISBN 0-375-82503-7)(Bright and Early Playtime Books)
• Sam and Gus Light Up the Night (ISBN 0-375-82926-1)(Bright and Early Playtime Books)
• My Nest Is Best (ISBN 0-375-83267-X)(Bright and Early Playtime Books)
• The Alphabet Book" (ISBN 0-375-80603-2)(Bright and Early Board Books)
• Big Dog...Little Dog: A Bedtime Story (ISBN 394-83312-0)(Pictureback Series)

He was the illustrator for:

• A Fish Out of Water, by Helen Palmer Geisel (ISBN 0-394-80023-0)(Beginner Book #23)
• Robert the Rose Horse, by Joan Heilbroner (ISBN 0-394-80025-7)(Beginner Book #25)
• I'll Teach My Dog 100 Words, by Michael K. Frith (ISBN 0-394-82692-2)(Bright and Early Books #17))
• I'll Teach My Dog a Lot of Words, by Michael K. Frith (ISBN 0-375-80099-9)(Bright and Early Board Books)

He wrote:

• Snow, illustrated by Roy McKie (ISBN 0-394-80027-3)(Beginner Book #27)

Eastman was a member of the American Civil Liberties Union, Westport Artists, the Screen Cartoonist Guild, and the Audubon Society.

Eastman was married in 1941 to Mary Louise Whitham of Glendale, California. He had two sons, Alan Eastman and Peter Anthony (Tony) Eastman, an animator, who himself directed video adaptations of Richard Scarry's children's books. Peter has also written and illustrated "Fred And Ted Go Camping"(2005), "Fred And Ted Like To Fly" (2007) and contributed new illustrations to several reissues of his father's books, "The Alphabet Book"(2000), "Big Dog... Little Dog" (2003), "Red Stop! Green Go!" (2004), and The Cat In The Hat Beginner Book Dictionary" (2007).