The Titans went back in front going into the end of the first, as Chris Johnson ended a three-yard run past the defense for the touchdown.
However, the hosts brought themselves within touching distance early in the second quarter when Matt Prater kicked a 25-yard field goal that was declared good.
Greene, however, was on hand again in the second as he completed a one-yarder to bring the score to 21-10, before Julius Thomas latched onto an eight-yard pass from Manning to ensure that the Broncos were still in with a shout.
The Broncos pulled within touching distance just before the break as Prater completed a 64-yard field goal to leave the score at 21-20 going into half time.
Their lead was extended to 13 with a simple one-yarder from Knowshon Moreno and an extra from Prater's kick.
However, a 41-yard pass from Ryan Fitzpatrick reached Justin Hunter as the Titans began to make inroads on the Broncos' advantage.
Fox's outfit were too strong, though, and Prater kicked a 19-yard field goal before Eric Decker took them to 44 after another touchdown pass from Manning.
Montee Ball made a smart five-yard run near the end of the fourth to wrap up victory for the Broncos.