First, laying out the facts:
Possessions (FGA + 0.5FTA + TO - OREB)- Heat 85, Bulls 87
Note: Individual possessions = FGA + 0.5FTA + TO - 1/3FGmisses
Points- Heat 82, Bulls 103
Points per possession- Heat 0.96, Bulls 1.18
Offensive rebounds- Heat 6, Bulls 19
Turnovers- Heat 16, Bulls 9
The Bulls' advantage in offensive rebounds and turnovers resulted in TWENTY "extra" possessions in Game 1. To put this in perspective, the difference between the best and worst offensive rebounding teams this year was about 5.5 per game. The difference between the best and worst turnover teams this year was about 4 per game. So if you take the best offensive rebounding team, combine it with the best team at limiting turnovers and combined the worst two teams in those categories, the difference would be about 9.5 "extra" possessions per game. The Bulls had more than double that many "extra" possessions against the Heat.
Multiply those 20 extra possessions by the Bulls 1.18 points per possession and you get 23.6 points. The Bulls won by 21, so it's quite easy to see why they won by the disparity that they did.
A lot of credit in Game 1 should go to Joakim Noah. He was poor shooting the ball, but managed to grab 8 offensive rebounds and only turned it over once. Luol Deng also quietly had a great game as usual. He used about 15 possessions while scoring 21 points. Additionally, he grabbed 2 offensive rebounds, 5 defensive rebonds, and had 4 steals. He drew the defensive assignment of LeBron James and performed admirably.
LeBron used approximately 18 possessions scoring only 15 points. Similarly, Dwyane Wade used approximately 20 possessions scoring only 18 points. Collectively, they accounted for only one offensive rebound (by LeBron).
Simply put, the Heat have no chance if LeBron and Wade account for less than one point per possession in this series. Add on top of that twenty extra possessions for the Bulls, and it shouldn't be surprising that the Bulls ran away with Game 1.
Can we expect this to continue the entire series?
The Bulls are certainly a great offensive rebounding team, but the Heat were the fourth best team in the league this year in terms of defensive rebounding percentage. The Heat and Bulls are fairly even in terms of turnover percentage. So, the short answer is no. We shouldn't expect that to happen the entire series. Though, if Tom Thibodeau saw something on tape that he thinks he can exploit, maybe it will continue.
It may be more likely, believe it or not, to expect LeBron and Wade to struggle offensively in this series. The Bulls' team defense is outstanding, and they did an excellent job keeping them out of the paint in Game 1. If you watch closely, every time LeBron and Wade start to penetrate, the Bulls put up a wall of defenders close to 12 feet out. Based on the Bulls' philosophy in Game 1, LeBron and Wade may have to shoot very well in order for Miami to win the series.