Bjorn and Streelman also helped their countries to the top of the leaderboard with five-under 66s at Royal Melbourne Golf Club.
South Korean KJ Choi, Scot Martin Laird and Welshman Stuart Manley were tied for third, a shot clear of Australian Jason Day.
The opening day left Denmark and defending champions the United States three shots clear atop the team standings from third-placed Portugal.
Sheffield’s Danny Willett is only three shots adrift in the individual standings after carding a 69, but his partner Chris Wood shot 75, leaving England in a tie for 11th with Germany, Brazil and Philippines.
Bjorn combined seven birdies with a double bogey at the par-four fourth, when he four-putted on the tricky and fast greens.
Streelman was left to rue a poor finish as he went to the 16th tee at seven under, only to drop a shot at that hole and the par-four 18th.
His countryman Matt Kuchar finished with an even-par 71 to leave the USA well-placed after the opening round.
World No 2 Adam Scott, winner on his past two starts, saw his chances of victory dealt a huge blow at the par-four 12th.
The Australian lost a ball with his first drive and his next was deemed unplayable, eventually ending up with a quintuple bogey as part of a four-over 75.
Day’s three-under 68 salvaged something for the hosts, leaving them at one over in the team standings, in what was a remarkably gutsy effort.
The Australian decided to play the event despite learning earlier this week that eight relatives, including his grandmother, died in Typhoon Haiyan in the Philippines.
Portugal are at two under in outright third after Ricardo Santos (69) and Jose-Filipe Lima (71) produced impressive rounds.
Canada (David Hearn and Brad Fritsch), Scotland (Laird and Stephen Gallacher) and South Korea (Choi and Bae Sang-Moon) are the other nations under par, locked at one under.
Matthew Nixon eagled his penultimate hole to open up a one-shot lead after day one of the South African Open at Glendower Golf Club in Ekurhuleni.
Marco Crespi, who has only had his European Tour card for a week, looked as if he would get his time on the big stage off to the perfect start and top the leaderboard after the opening round until Nixon’s late heroics displaced him and South African Jbe Kruger at the the summit.
Crespi finished third at the Qualifying School at PGA Golf Catalunya last week to secure his place on Tour for the 2014 season and yesterday fired an opening 65.
He recorded four consecutive birdies from the 12th to the 15th to turn in 32 and picked up a further three shots on the back nine.
Kruger, meanwhile, eagled the eighth and then birdied five of his last seven holes.
Twice US Open champion Retief Goosen shot 66.