Norman had won just two of nine matches this season and needed a wild card into the main draw of an event he won in 1997 and 2000.
"Frankly, I'm just happy that my body is holding up for this level of tennis," Norman said. "I was a bit confused in the first set when Jonas was serving and volley. But then I played better and turned the match in my favor."
The eighth-seeded Bjorkman, who won his fifth career title in Nottingham last month, has lost in the first round of seven of his past eight tournaments. He received a hero's welcome here on Monday after winning the Wimbledon men's doubles title with Australia's Todd Woodbridge.
Norman beat Bjorkman for the sixth time in seven meetings and will play Slovakia's Dominik Hrbaty in the second round. Hrbaty beat Arnaud Di Pasquale of France, 6-3, 6-4, on Tuesday.
Top seed Guillermo Canas of Argentina improved his match record this season to 32-16 with a 6-4, 6-2 victory over Sweden's Andreas Vinciguerra.
Third-seeded Carlos Moya and fourth-seeded Tommy Robredo of Spain advanced to the second round on Tuesday. In his first ATP match in a month, Moya defeated Olivier Rochus of Belgium, 6-3, 6-4, and Robredo rallied to beat Kristian Pless of Denmark, 5-7, 6-2, 6-2.
Defending champion Andrea Gaudenzi of Italy was beaten by Gorka Fraile of Spain, 6-4, 7-6 (7-4).
In other matches, Nicolas Coutelot of France downed Filip Prpic of Sweden, 3-6, 7-5, 6-2 and Irakli Labadze of Georgia defeated Victor Hanescu of Romania, 6-4, 1-6, 6-3.
Swim Week Miami Beach 2014 [PHOTOS]