Cox, a £300,000 summer arrival from Reading, started the season how he finished the last by terrorising a hapless Tranmere defence who had manager Ronnie Moore fuming at the end of the game.
Strike partner Billy Paynter opened the scoring on five minutes and his goal owed as much to the determination of his fellow forward as it did to the fine headed finish he eventually produced.
Cox could have settled for a corner as the ball rebounded along the touchline, but the lively striker stretched to keep it in and allow Jon-Paul McGovern to cross for Paynter to tower above the Tranmere defence.
Swindon dominated thereafter but were fortunate not to concede an equaliser when they momentarily took their eye off Tranmere midfielder Charlie Barnett as the former Liverpool trainee burst into the area unmarked.
Barnett controlled well with his chest but, as he attempted to hook a volley over Petr Brezovan, he miscued horribly.
The away side were then made to pay for not taking that chance as Brezovan's ensuing goal kick was helped on by Paynter for Cox to finish convincingly, striking well past John Achterberg.
The lively Anthony McNamee nearly capped a fine display with a goal on the stroke of half-time as he nipped inside on to his right foot with a drop of the shoulder and a step-over but Achterberg was his equal, saving well at the near post.
Tranmere's respite lasted only until the 51st minute as the hosts gave themselves an unassailable lead.
Cox once more was the catalyst as he found space on the left-hand side of midfield before driving a cross into the path of McGovern to smash in a third from six yards.
Swindon could and should have made it four on the hour-mark as Cox was left unmarked and in space, but this time the forward was not so incisive with his passing as he attempted to set up Lilian Nalis but his centre was poor.
Swindon lost momentum in the final ten minutes and allowed Tranmere a late consolation.
Substitute Edrissa Sonko had a shot blocked by the sprawling Jerel Ifil, but he regained his composure to slide a pass into Bas Savage to tap in from close range for a goal on his Tranmere debut.