After a scoreless first, the Flames lit the Islanders up for four goals in the second.
On the very next shift Monahan was called for interference and on the ensuing power play New York captain Tavares banged in a rebound to tie the game 1-1.
It was all Calgary after that as the Flames responded with three straight goals.
Stajan got the go-ahead goal when he snuck the puck through a crowd and under Nabokov, who was replaced a few shifts later by Ken Reiter.
Backlund gave Calgary a two-goal lead when he scored on his own rebound and Monahan buried a loose puck in front to make it 4-1 after two periods.
Matt Moulson added one more for New York early in the third on a nice set up from Tavares.
Calgary goaltender Karri Ramo made 21 saves on the night, two of which were sensational.
Nabokov took the loss for the Islanders.
The matchup marked the first pre-season game for the Islanders and the fourth for the Flames, who lost to 4-2 to the Ottawa Senators Monday night in Saskatoon 4-2.
The last time a pre-season NHL game was played in Regina was 2009 when Ottawa beat the Tampa Bay Lightning 3-1 in front of 6,450 fans.
Not nearly that many showed up for this game. The attendance was never announced but the managers of the Brandt Centre told reporters 4,189 tickets were sold.