Bjorn and Streelman helped their countries to the top of the leaderboard with five-under 66s at Royal Melbourne Golf Club on Thursday.
South Korean KJ Choi, Scot Martin Laird and Welshman Stuart Manley are 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.
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 number two 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 decent 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.