There is research into this topic but it's not proven.
There's too much variability to consider.
Take the '04 Detroit Pistons (it's a City, I know, but the premise scales). In 2004 the Detroit Pistons were underdogs to the Los Angeles Lakers. Economically, LA was in a much better place than Detroit but Detroit scrapped it's way to a victory (and a return to the finals but a loss the next year to the San Antonio Spurs (Go Spurs Go)). Detroit's victory may have led to sales growth in player's jerseys (Richard Hamilton and Ben Wallace for instance) but the sports victory did not save the city from it's demise and subsequent bankruptcy.
I believe the economic benefits of professional sports victories are more cosmetic than practical. A real estate agent may have a few more selling points in an economic transaction but the gains are usually temporary as it's hard to have back-to-back victories in professional sports...
Controversially, the theory MAY ACTUALLY apply to collegiate sports and the economic prowess of a university. Ever been to University of Georgia or Alabama? Those walls aren't lined with money from their journalism schools, I can tell you that much.
It'd be hard to prove that professional sports wins tie to economic success given how many other exogenous factors there are to economic success but there's data available to analyze this question (specifically, GDP growth and World Cup/Olympic Cup/World Title Events).
I don't know if governments are tossing up grant money for this research though.