The yield on the 10-year Treasury soared to as high as 2.43 percent following the release, about seven basis points higher from where it was ahead of the release. It closed at 2.402 percent, the highest settlement since October 6.
Yields on 30-year Treasurys rose to 3.1366 percent from about 3.06 percent. It last held at 3.11 percent.
The yields on 2-year Treasury notes also spiked from 0.67 percent to 0.79 percent. It was last up at 0.71 percent.
"Bottom line, the very good job gain in May puts the 6 month average at 236k vs the 260k average in 2014 as the May gain made up from only modest job gains in January, March and April," Peter Boockvar, chief market analyst at The Lindsey Group, said in a note.
Treasury yields have tracked German Bund yields sharply higher this week amid a view that bond-buying by the European Central Bank may have helped avert the risk of deflation. The ECB this week raised its inflation forecasts while data on Tuesday showed inflation in the euro zone rose by more than expected in May.
"Combining a still good growth rate of over 200k per month with the uptick in average hourly earnings is making the Fed's zero rate policy looking more and more inappropriate to say the least and the US Treasury market is not waiting around for the Fed to act," he added.