Bledsoe's six-footer high off the glass put the Suns up 97-93. It came after a 7-0 Chicago run that cut the lead to 95-93 on Aaron Brooks' layup with 31.9 seconds to go.
Markieff Morris had 12 points and 14 rebounds for the Suns. His twin brother, Marcus, also scored 12.
Derrick Rose scored 23, Jimmy Butler, 22, and Brooks, 21 for the Bulls, who were coming off a double-overtime loss to the Lakers in Los Angeles Thursday night and didn't arrive in Phoenix until 3:30 a.m. on Friday.
The Suns finished 6-2 in a franchise-record eight-game homestand.
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