Tekulve is a 1969 graduate of Marietta College in Ohio. He signed that year as a free agent with the Pittsburgh Pirates and remained with that organization until 1985. He made his major league debut in 1974.
His best seasons came in 1978 and 1979, in both of which he saved 31 games and posted ERAs of 2.33 and 2.75, respectively. He saved three games in the 1979 World Series including the winner, as his Pirates defeated the Baltimore Orioles. He was selected an All-Star in 1980.
Early in the 1985 season, Tekulve was traded to the Philadelphia Phillies for Al Holland and a minor leaguer, Frankie Griffin. He continued to be an effective reliever into his 40s. Only in his first season (1974) and his last season (1989) did he post an ERA above 4. While with the Phillies, he led the NL in games pitched for the fourth time, with 90 in 1987 at the age of 40.
Tekulve signed with the Cincinnati Reds before the 1989 season and pitched in 37 games before retiring in July.
Tekulve led the major leagues in games pitched four times, appearing in 90 or more games three times. He and Mike Marshall are the only pitchers in baseball history to appear in 90 or more games more than once (each did it three times). Tekulve is also the oldest pitcher ever to appear in 90 games, when he did so in 1987 at age 40. Tekulve's three saves in the 1979 World Series tied the single-Series mark set by Roy Face in the 1960 World Series; it was broken by John Wetteland in 1996. He holds the National League record for career innings pitched in relief (1,436⅔), and formerly held the major league record for career relief appearances; his 1,050 career games, all in relief, ranked second in major league history to Hoyt Wilhelm's 1,070 when he retired. Tekulve owns the career records for most appearances and innings pitched without making a single start. In 1986 he broke Roy Face's NL record of 846 career games pitched; he held the record until John Franco passed him in 2004. In August of 1987, he pitched on nine consecutive days, a record for pitchers.
Tekulve also holds the record for most career losses without having given up any earned runs, with 12, as well as the record for most intentional walks issued, with 179.
Tekulve appeared in a 1983 episode of Mister Rogers' Neighborhood to explain how people play baseball.
Tekulve was a member of the Philadelphia Phillies television broadcast team from 1991 to 1997.
After several years involvement with the Washington Wild Things of the independent Frontier League, Tekulve took a job as the Pittsburgh Pirates' advance scout in 2006.
Tekulve currently works for AT&T SportsNet Pittsburgh (formerly FSN Pittsburgh and later Root Sports Pittsburgh) and appears as an analyst after each Pittsburgh Pirates game since 2008. However, in the early to mid part of the 2014 Pirates season, he took a hiatus for personal reasons. Filling in for him in his absence, former Pirate player and former Oakland Athletics manager Ken Macha.
Tekulve underwent successful heart transplantation surgery on September 5, 2014 after spending eight months on the transplant list. The surgery was performed at Allegheny General Hospital.
Tekulve threw out the ceremonial first pitch at the National League Wild Card game between the Pittsburgh Pirates and the San Francisco Giants on October 1, 2014.
Tekulve announced his retirement from broadcasting on September 5, 2017, after the Pirates' 4-3 win over the visiting Chicago Cubs.