Travis Zajac, Zach Parise and Brian Rolston also lit the lamp for the Devils, who snapped a two-game slide and have won five of seven. Jamie Langenbrunner and Patrik Elias picked up two assists each and Scott Clemmensen made 23 saves.
Clarkson capitalized on a turnover at the Sabres blue line, skating in alone and sliding a shot through Miller 6:04 into the third period for a two-goal Devils lead.
Rolston made it 5-2 at 9:31, following Clarkson's initial shot, corralling the loose puck at the side of the crease and lifting it over a fallen Miller.
However, Gaustad's put-back of Jason Pominville's right-circle blast 12 seconds later brought the Sabres within a pair.
Though the Sabres pulled Miller with more than a minute remaining plus enjoyed a 6-on-4 advantage after Devils backliner Colin White went off for interference, New Jersey held on for the win.
Zajac's wrister from the slot put the Devils on the board 1:36 into the game, but Stafford tipped home Clarke MacArthur's hard cross-ice pass for a 1-1 tie at 6:58.
Parise completed a 5-on-3 power play for a 2-1 lead with 8:16 to play, stashing home his own rebound after Langenbrunner passed from the left point to the young sniper in the right circle.
Bryce Salvador's seeing-eye wrister less than a minute after the midway point of the second period gave New Jersey a 3-1 edge, but Kaleta responded for Buffalo 18 seconds later.