Peacock headed a Michael Timlin free-kick home seven minutes after half-time to add to Lucas Jutkiewicz's opener in the first half.
Wycombe, the better side throughout much of the game, snatched a consolation through Jermaine Easter but didn't have enough to come away with a point.
Following their nightmare display against relegation-threatened Macclesfield three days ago, Paul Sturrock's side nearly got off to a similarly disappointing start when they fell asleep at a throw-in.
Russell Martin's throw found Kevin Betsy in oceans of space inside the penalty area on 14 minutes, but the Wycombe winger hit his shot straight at Phil Smith.
Within a minute Christian Roberts tried to test Jamie Young in the Wycombe goal with a low drive but the former Reading trainee made a comfortable save.
As the minutes ticked towards half-time Swindon took the initiative when 17-year old Jutkiewicz scored his third goal in four games.
Roberts rounded Stephen O'Halloran at left-back and clipped a far-post cross on to the head of the teenage striker for him to head into the top corner from five yards.
No sooner had they grabbed the first goal than Swindon helped themselves to a second eight minutes after the break when the robust Jutkiewicz was brought down on the corner of the penalty area.
On-loan midfielder Timlin sent a peach of a cross into the area and Peacock wasted no time in glancing his header into the corner of the net for his seventh of the season.
Shell-shocked Wycombe continued to press, full of possession but without any penetration.
Full-back Martin tried a long-range dipping half-volley but it was straight at Smith before Mike Williamson missed from close range.
The away side finally got their goal when Betsy laid the ball into Easter's path for him to smash a consolation from close range but it proved too little too late.