The son of tennis coach and mailman Lars Björkman, Jonas began playing tennis at the age of six. At 18, he won the Swedish Junior Championship and was among the top 5 junior Swede players. He married Petra on 2 December 2000 in Stockholm and has a son, Max (born 15 January 2003). He plays right-handed and has a particularly good record against left-handed players. He claims it's because his father plays left-handed.
He turned professional in 1991. In 1993, he won three Challenger singles titles. In 1994, he won seven titles in doubles including the 1994 ATP Tour World Championships in Jakarta. In 1995, he reached his first career ATP singles final in Hong Kong. In 1997, he became the 9th ever Swedish tennis player to finish in ATP top 10 at no. 4. He advanced to his first Grand Slam semifinal at the US Open, defeating Francisco Clavet, Todd Martin, Gustavo Kuerten, Scott Draper and Petr Korda before losing to Greg Rusedski. At the 1998 Australian Open, he won his first career doubles Grand Slam title. 2000 saw him finishing in the singles top 50 for the fifth time in seven years.
In his ATP career, he has won six singles titles and 54 titles in doubles, including nine Grand Slam titles in doubles.
He made his Davis Cup debut in 1994 and has played regularly for Sweden ever since. He has compiled a 21–14 record in doubles and a 14–9 record in live singles rubbers. He was a member of Sweden's Davis Cup championship teams in 1994, 1997, and 1998.
In 2002, Björkman won the Nottingham Open by defeating Wayne Arthurs in the final, however at Wimbledon he found himself drawn against top seed Lleyton Hewitt in the first round. Björkman was defeated in straight sets, as Hewitt went on to win the tournament.
In the 2006 Wimbledon, he unexpectedly made it into the singles semi-finals at the age of 34, making him the oldest player to get there since Jimmy Connors in 1987. He had only made it into the singles quarter-finals once in 2003. He was unseeded, but defeated 14th-seeded Radek Štěpánek in a match which included saving a match point. He had previously ousted his doubles partner Max Mirnyi and another Swede, Thomas Johansson, to make the quarter-finals. In the semi-final he found World No. 1 and defending champion Roger Federer too good and was overpowered in straight sets, 6–2, 6–0, 6–2. When John McEnroe announced his official return to the ATP Pro Tour in 2006 he teamed up with Björkman to win the doubles title at the SAP Open in San Jose.
During Wimbledon in 2008, he announced that he would be playing in his final Wimbledon as he was planning on retiring at the end of the season. Although being knocked out in the first round of singles, Björkman and Kevin Ullyett made it to the final, being defeated by second seeds Daniel Nestor and Nenad Zimonjić, though receiving a hero's farewell to his extensive career at Wimbledon.
Björkman retired from professional tennis after the Swede and his partner Kevin Ullyett failed to qualify for the doubles semifinal at 2008 Tennis Masters Cup - Doubles.
On 2 October 2013, he announced the comeback on tour in If Stockholm Open doubles draw, receiving a wild card in pair with fellow countryman Robert Lindstedt.