Engel was born in Washington, D.C., the daughter of Ruth Caroline (née Hendron) and Benjamin Franklin Engel, who was a Coast Guard admiral. Engel attended the Kodiak Island Borough School District, and Walter Johnson High School and the Academy of the Washington Ballet from which she graduated. She earned her college degree from the University of Hawaii at Manoa.
After college, Engel appeared in musical productions with Washington's American Light Opera Company. She moved to New York City in 1969, appearing Off-Broadway in Lend an Ear, and for a year as Minnie Fay in the Broadway production of Hello, Dolly!, starting in December 1969. A 1971 off-Broadway production of The House of Blue Leaves eventually played in Los Angeles, where Engel was seen by Mary Tyler Moore and her husband, producer Grant Tinker, her soon-to-be employers.
Engel appeared as Georgette Franklin Baxter on The Mary Tyler Moore Show from 1972 until the sitcom ended in 1977. The role won her two Emmy nominations. After that sitcom ended, she teamed up with former Mary Tyler Moore Show co-star Betty White for The Betty White Show during its first and only (1977–78) season. She later co-starred on two short-lived 1980 sitcoms, Goodtime Girls as Loretta Smoot, and in Jennifer Slept Here featuring Ann Jillian.
Engel had a recurring role on Coach as Shirley Burleigh and starred as the voice of Love-a-Lot Bear in The Care Bears Movie (1985). She played a good witch in a 2007 recurring role of Esmeralda on the now-defunct NBC soap opera Passions. Engel received consecutive Emmy Award nominations as outstanding guest actress in a comedy series in 2003, 2004 and 2005 for her role on Everybody Loves Raymond as Robert Barone's mother-in-law, Pat MacDougall.
While her movie appearances have been sporadic, Engel made her film debut in Miloš Forman's first English-language movie Taking Off for which she was nominated for a British Academy Award for best supporting actress. Other film appearances include The Outside Man (1973), Signs of Life (1989),  Papa Was a Preacher (1987), The Sweetest Thing (2002) and the made-for-TV movies The Day the Women Got Even (1980) and A Love Affair: The Eleanor and Lou Gehrig Story (1978).
Engel returned to her stage roots in 2006, appearing on Broadway in the musical The Drowsy Chaperone, with Sutton Foster and Edward Hibbert. She created the role of Mrs. Tottendale, which she continued to perform, leaving the Broadway production as of April 1, 2007. She was featured in the North American tour, performing in Toronto in September 2007, through engagements at the Orpheum Theater in San Francisco, in August 2008, and at the Denver Performing Arts Complex in October 2008.
For the summers of 2004, 2005, 2007, 2009 and 2010, Engel appeared in various productions at The Muny Theater in Forest Park in St. Louis, Missouri. She most recently appeared in Show Boat in August 2010 as Parthy. In July 2005, she appeared in Mame as Agnes Gooch, in June 2007 she appeared in Oklahoma! as Aunt Eller, and in July 2009 she appeared as Mrs. Paroo in The Music Man.
In June 2010, Engel appeared at the Ogunquit Playhouse in Ogunquit, Maine production of The Drowsy Chaperone as Mrs. Tottendale. In October through December 2010, Engel was featured in the Vineyard Theatre's Off-Broadway production of Middletown, written by Will Eno.
In 2012, she appeared in episodes of The Office as an older lady being helped by Erin Hannon (Ellie Kemper) and in the episode called "Palmdale, Ech" of Two and a Half Men as the mother of Lyndsey MacElroy portrayed by Courtney Thorne-Smith. In March 2012, 35 years after the close of The Mary Tyler Moore Show, Engel was reunited with Betty White in the third season of Hot in Cleveland as Mamie Sue Johnson, best friend of White's character Elka, in a continuing, recurring role.
Engel appeared in the new Annie Baker play John, which opened Off-Broadway at the Signature Theatre on July 22, 2015 (previews), directed by Sam Gold. The play ran to September 6, 2015. The cast also features Lois Smith. Engel won a 2016 Obie Award for Distinguished Performance by an Actress and was nominated for the 2016 Lucille Lortel Award, Outstanding Lead Actress in a Play for her role in this play.
Engel starred in the new musical Gotta Dance, which premiered at the Bank of America Theatre, Chicago on December 13, 2015, running through January 2016. The cast also stars Stefanie Powers, Lillias White and Andre DeShields. The musical is directed and choreographed by Jerry Mitchell, with a book by Chad Beguelin and Bob Martin, and the score by Matthew Sklar and Nell Benjamin.
|1971||Taking Off||Margot||Nominated — BAFTA Award for Best Actress in a Supporting Role|
|1972||The Outside Man||Mrs. Barnes|
|1978||A Love Affair: The Eleanor and Lou Gehrig Story||Claire Ruth|
|1980||The Day the Women Got Even||Kathy Scott|
|1983||The Magic of Herself the Elf||Willow Song||voice|
|1985||Papa Was a Preacher||'Mama' Porter|
|1985||The Care Bears Movie||Love-a-Lot Bear||voice|
|1989||Signs of Life||Betty|
|2001||Dr. Dolittle 2||Giraffe||voice|
|2002||The Sweetest Thing||Vera|
|2006||Boog and Elliot's Midnight Bun Run||Bobbie||voice|
|2007||Nunsensations||Sr. Marie Eugene|
|2008||Open Season 2||Bobbie||voice|
|2010||Open Season 3||Bobbie||voice|
|2013||Grown Ups 2||Mrs. Lamonsoff|
|1974||Rhoda||Georgette Franklin||2 episodes: Rhoda's Wedding (Parts 1 & 2)|
|1972–77||The Mary Tyler Moore Show||Georgette Franklin, later Georgette Franklin Baxter||56 episodes
Nominated — Primetime Emmy Award for Outstanding Supporting Actress in a Comedy Series (1976–77)
|1977–78||The Betty White Show||Mitzi Maloney||14 episodes|
|1979||Mork & Mindy||Ambrosia Malspar||2 episodes: The Exidor Affair and Exidor's Wedding|
|1980||Goodtime Girls||Loretta Smoot||13 episodes|
|1977–82||The Love Boat||Cleo Bagby||4 episodes|
|1978–83||Fantasy Island||Brenda Rappaport||5 episodes|
|1983–84||Jennifer Slept Here||Susan Elliot||13 episodes|
|1992||Hi Honey, I'm Home!||Georgette Franklin Baxter||1 episode: Elaine Takes a Wife|
|1991–97||Coach||Shirley Burleigh||17 episodes|
|2003–05||Everybody Loves Raymond||Pat MacDougall||13 episodes
Prism Award for Best Performance in a Comedy Series (2006)
Nominated — Primetime Emmy Award for Outstanding Guest Actress in a Comedy Series (2003–05)
|2012||The Office||Irene||3 episodes|
|2012||Two and a Half Men||Jean||2 episodes: Palmdale, Ech and Grandma's Pie|
|2012–15||Hot in Cleveland||Mamie Sue Johnson||18 episodes|
|2018||One Day at a Time||Sister Barbara||Episode: Homecoming|