Do immigrants win more Nobels than native born Americans? That depends who you consider an immigrant. In 2016 eleven people won the Nobel prize. Seven of them have spent a considerable amount of time in the United States, though not all of them would label themselves Americans. Of those seven, one was born in Finland, five were born in England and one is a native born American.
The images bellow are screenshots from a workbook hosted on Tableau Public. If you would like to play with the interactive versions of these visualizations or download the underlying data (which was collected from Wikipedia) please go to Tableau Public.

US Nobel Laureates since 2010. Blue indicates the laureate is native born. Foreign born laureates are colored orange.
That’s a large percentage of Nobels going to immigrants, but 2016 is an anomaly. Since 2010, 10 of the 33 Americans who have won Nobels have been foreign born. While the percentage of Nobels going to immigrants varies a lot by year, the percentage is remarkably consistent if you look at it by decade. This percentage has hovered between 22 and and 30 percent for the past eight decades. That is higher than the percentage of immigrants among the general American population (which is has bounced between 4% and 15% over the past century and currently stands at 13%). I couldn’t find the percentage of American academics who are foreign born. My experience in grad school leads me to believe it is higher than the percentage in the general population, so I don’t know if immigrants are truly overrepresented among Nobel laureates.
Overall 25 percent of American Nobel laureates have been immigrants since the prize was founded. This is a bit above average among countries which have won more than 10 Nobels and the highest percentage of any country which has won 15 or more Nobels. Some may take this as an argument that relaxing immigration policies will lead to more Nobel laureates, but that argument is shaky at best. The correlation is positive but it is not significant.

The correlation line shown has a p value of .29
A more down to earth question for those of us who don’t control immigration policy is, based on your country of birth, should you move if you want to win a Nobel (assuming the distribution will follow historical data). If you call western Europe, Russia, Australia or the US home the answer is decidedly no. However for my friends in China, Eastern Europe and Canada this might be a good strategy (assuming you already have the being crazy smart thing covered). If you do choose to move the best country to move to is probably the US or the UK.

Countries which are colored dark red had no citizens who moved elsewhere win a Nobel. Dark blue indicates that all Nobel Laureates born in that country emigrated.
It might pay to check out the data specific to your subfield before you apply for your visa though. A lot of chemistry and biology Nobel laureates have been immigrants but those who win for peace and literature usually stay home. In 2016 the native born American won for literature while the prizes raked in by immigrants covered physics, chemistry and economics. Historically female Nobel laureates have been more likely to be immigrants than their male counterparts but if you look at the percentages over time that trend has reversed itself. Since 2000 23% of male Nobel laureates have been immigrants while only 17% of female Nobel laureates have been immigrants.
The Nobel prize is not the only measurement of success. Most laureates in the sciences are associated with large universities in rich countries which have the resources to help them get their best work done. The research that happens at these institutions transforms the world we live in and saves lives. Having that kind of impact on the world is a much bigger prize than the Nobel.