Cleveland made a bright start and scored their first touchdown midway through the first quarter, with Campbell picking out Davone Bees in the end zone to finish an 80-yard drive.
Baltimore chipped away at that early lead with a field goal but were undone again by the combination of Bees and Campbell at the start of the second quarter, falling 14-3 behind.
When play resumed, Cleveland were able to put Flacco under consistent pressure to retrieve possession, allowing Gary Barnidge to extend their advantage in the third quarter.
Back come Baltimore once again, though, as they moved to within three points early in the fourth quarter, with Brown once again finding space in the end zone before making a two-point conversion.
Cleveland stayed calm to close out the match late on as Campbell killed the clock in a drive lasting 18 plays, before Billy Cundiff kicked over a 22-yard field goal in the final 20 seconds.