Byrne saw his side hang on and survive Leyton Orient's late blitz as Tamika Mkandawire's goal was too little too late.
Wilson could not have scripted a better start from his new side as the game was just 42 seconds old when the visitors took the lead.
John-Paul McGovern played the ball in from the left and Peacock, on the edge of the box, crashed the ball into the roof of the net in spectacular style.
Orient responded and should have been on level terms within four minutes when Adam Boyd carved out an opening for Sam Parkin and his pass sent Adam Chambers clear, but the Orient midfielder rolled the ball wide of target.
Orient's top-scorer, Boyd, then headed wide, but the visitors always looked dangerous on the break and home goalkeeper Glenn Morris had to dive full-length to keep out a glorious effort from Simon Cox.
After the interval, Orient started strongly and Boyd fired just over before the game changed by a controversial penalty award on the hour.
Referee Jarnail Singh pointed to the spot on the advice of his assistant as he penalised Alton Thelwell for a harsh handball.
Swindon defender Smith rolled the ball neatly to the left of the home goalkeeper to give the visitors a 2-0 lead.
Orient threw caution to the wind in the closing stages and played with just three at the back and almost pulled a goal back on 77 minutes, but Town goalkeeper Petr Brezovan made a stunning stop from home substitute Ryan Jarvis.
Orient finally cut the deficit on 79 minutes when Boyd headed the ball back across goal for Mkandawire to head home from close range and in a drama-drenched ending Jarvis, Mkandawire, Boyd and Parkin all had chances to level, but gritty Swindon held on.