The Saints were without star running back Alvin Kamara against their NFC South rivals due to a rib injury, but Winston was cleared to play.
The former Bucs quarterback was known to have a back injury, and the severe nature of the issue was reported ahead of kickoff.
FOX Sports' Jay Glazer revealed the four fractures, saying Winston was "dealing with an awful lot of pain" but had "no risk of further damage".
Winston was influential in the Saints' 27-26 Week 1 win over the Atlanta Falcons, throwing for 269 yards and two touchdowns.
Both scores and 213 yards came in the fourth quarter, with Winston the first Saints QB to have 200 fourth-quarter passing yards in a game since Drew Brees' 294, also against the Falcons, in Week 10 in 2008.