Nacho Monreal had put the Gunners ahead when he met a curling ball from Mesut Ozil.
But within 60 seconds, Swansea were level as Alfie Mawson's pass bisected the Arsenal defence and Clucas ran on to score.
A minute after Henrikh Mkhitaryan came on for his Arsenal debut, Jordan Ayew put the Swans ahead as Petr Cech miskicked a backpass from Shkodran Mustafi straight into his path.
Ayew then set up Clucas to seal Swansea's second successive victory.
The win lifts Swansea from 20th to 17th. Arsenal lie sixth, eight points off fourth place.
Swansea: 3 Feb v Leicester (A)
Arsenal: 3 Feb v Everton (H)