Born in Bonham, Texas, and raised in Oakland, California, Morgan was nicknamed "Little Joe" for his diminutive 5'7" stature. He was a standout at Castlemont High School before being signed by the Houston Colt .45s as an amateur free agent in 1962.
Morgan played ten seasons for Houston, compiling 72 home runs and 219 stolen bases. He made the All Star Team twice during this period, in 1966 and 1970. On June 25, 1966, Morgan was struck on the kneecap by a line drive (hit by Lee May) during batting practice. The broken kneecap forced Morgan out of the lineup for 40 games, during which the Astros went 11-29 (for a .275 winning percentage).
Although Morgan played with distinction for Houston, the Astros wanted more power in their lineup. Additionally, manager Harry Walker considered Morgan a troublemaker. As a result, they traded Morgan to the Cincinnati Reds as part of a blockbuster multi-player deal on November 29, 1971, announced at baseball's winter meetings.
After joining The Big Red Machine, Morgan's career reached a new level. This included eight consecutive All-Star Game appearances (1972–79) to go along with his 1966 and 1970 appearances with Houston.
Morgan (especially towards the end of his work with ESPN) was accused of often not doing his homework prior to broadcasts and seeing no team as performing better than his own 1975–76 Cincinnati Reds teams. He was also criticized for coming across as dry and humorless, while making statements perceived as aloof and distant, despite his partner Jon Miller's efforts to draw him into conversations.
Morgan returned to Houston in 1980 to help the young Astros win the NL West. The Astros then lost the National League Championship Series to the Philadelphia Phillies.
Morgan went to the San Francisco Giants for the next two seasons. His home run in the last game of the 1982 season eliminated the Dodgers from the division race. He won the 1982 Willie Mac Award for his spirit and leadership.
From 1985 to 1988 Morgan called college baseball games for ESPN. In 1989, Morgan teamed with Brent Musburger to call the championship game of the College World Series for CBS.
Morgan had spent a previous stint (from 1986 to 1987) with NBC calling regional Game of the Week telecasts alongside Bob Carpenter. During NBC's coverage of the 1985 and 1987 National League Championship Series, Morgan served as a pregame analyst alongside hosts Dick Enberg (in 1985) and Marv Albert (in 1987).
From 1988 to 1989 Morgan served as an announcer for ABC, where he helped announce Monday Night and Thursday Night Baseball games (providing backup for the lead announcing crew composed of Al Michaels, Tim McCarver and Jim Palmer), the 1988 American League Championship Series with Gary Bender and Reggie Jackson, and served as a field reporter for the 1989 World Series along with Gary Thorne (Morgan's regular season partner in 1989). Morgan was on the field at San Francisco's Candlestick Park alongside Hall of Famer Willie Mays (whom Morgan was getting set to interview) the moment the Loma Prieta earthquake hit at 5:04 pm.
Morgan was a member of ESPN's lead baseball broadcast team alongside Jon Miller and Orel Hershiser. Besides teaming with Miller for Sunday Night Baseball (since its inception in 1990) telecasts, Morgan also teamed with Miller for League Championship Series and World Series broadcasts on ESPN Radio.
From 1994 to 2000 Morgan teamed with Bob Costas and Bob Uecker (until 1997) to call baseball games on NBC (and in association with The Baseball Network from 1994 to 1995). During this period Morgan helped call three World Series (1995, 1997, and 1999) and four All-Star Games (1994, 1996, 1998, and 2000). Morgan also called three American League Championship Series (1996, 1998, and 2000) and three National League Championship Series (1995 alongside Greg Gumbel, 1997, and 1999).
In 1999, Morgan teamed with his then-NBC colleague Bob Costas to call two weekday night telecasts for ESPN. The first was on Wednesday, August 25 with Detroit Tigers playing against the Seattle Mariners. The second was on Tuesday, September 21 with the Atlanta Braves playing against the New York Mets.
In 2006, he called the Little League World Series Championship with Brent Musburger and Orel Hershiser on ABC, replacing the recently fired Harold Reynolds. During the 2006 MLB playoffs, the network had Morgan pull double duty by calling the first half of the Mets–Dodgers playoff game at Shea Stadium before traveling across town to call the Yankees–Tigers night game at Yankee Stadium.
In 2009, Sports Illustrated's Joe Posnanski spoke about the perceived disparity between Morgan's celebrated playing style and his on-air persona:
On April 21, 2010, it was announced that Morgan was returning to the Reds in the role of "special adviser to baseball operations." Morgan's role is to work in both baseball and community outreach for the Reds.
It was announced on June 17, 2011, that Morgan would begin a daily, one-hour general-sports-talk radio program, beginning August 22.
Morgan was an extremely capable hitter—especially in clutch situations. While his lifetime average was only .271, he hit between .288 and .327 during his peak years with the Reds. Additionally, he drew many walks, resulting in an excellent .392 on-base percentage. He also hit 268 home runs to go with 449 doubles and 96 triples, excellent power for a middle infielder of his era, and was considered by some the finest base stealer of his generation (689 steals at greater than 80% success rate). Besides his prowess at the plate and on the bases, Morgan was an exceptional infielder, winning the Gold Glove Award in consecutive years from 1973 to 1977.
"While I’m best known for baseball, I’ve always had a love of all sports", Morgan said in a statement. "I’m fortunate that my career has allowed me to meet some of the most amazing people, and I have heard so many remarkable stories. With my new show, I am looking forward to sharing these stories, as well as speaking with today's Sports personalities and newsmakers", Morgan concluded.