Nine wickets fell for 97 runs in 32 overs as Durham were bowled out for 149 in their second innings at Darlington, good slip catching backing up the bowlers.
Pigott, who was released by Sussex last season, had taken only one wicket in two previous Championship games for Surrey, but he finished with 4 for 22.
Cuffy removed the nightwatchman, David Graveney, in the day's second over. John Morris lasted four balls before he fell and Cuffy then uprooted Phil Bainbridge's off-stump.
Joey Benjamin trapped Stewart Hutton leg before and, when Mark Saxelby edged an attempted drive to slip in Pigott's first over, Durham had slipped from 52 for 1 overnight to to 91 for 6.
Mushtaq Ahmed bowled Somerset to a dramatic 53-run victory in his final Championship appearance of the season, at Taunton. The leg-spinner, who joins Pakistan's tour party to Sri Lanka at the weekend, took 7 for 94.
Worcestershire, chasing 332, reached 209 for 2, thanks to 83 from Philip Weston and 79 from Tom Moody, but three wickets then fell for four runs. Mushtaq ended the resistance of their last man, Richard Illingworth, caught behind with seven balls remaining.
Min Patel gave more substance to his claims for a place in the England team by spinning Kent to their first Championship victory of the season, by 175 runs over Yorkshire at Maidstone. Patel, 24 on Thursday, produced a spell of 3 for 14 in 12 balls to finish with 5 for 68.Reuse content