Two second-half goals from super-subs Christian Roberts and Billy Paynter were enough to earn the Robins a 2-1 victory.
Wycombe's proud Cup record - Carling Cup semi-finalists last season and FA Cup semi-finalists in 2001 - was dented despite a solid opening period.
Twenty-three league places separate the two teams but on the first-half showing the gap was far from obvious, if anything, Wycombe looked the superior side.
Wanderers' winger Martin Bullock caused Swindon problems on the right flank and almost created an opening goal with several testing crosses. Meanwhile Scott McGleish was thankful for an offside flag sparing his blushes after he spooned another chance over the crossbar.
Sturrock, who made several changes to his team for this fixture was forced to turn to his bench to rescue the Robins. Then substitute Roberts, on at half-time, made the breakthrough, capitalising on an error by Wycombe left-back Tom Williams and slotting past keeper Frank Fielding. It proved to be the turning point.
Wycombe, so combative up to this moment, were rattled and Paynter sealed the match with 17 minutes to go.
The striker, to whom Sturrock said he had given a 2wee spanking on the bum2 by relegating him to the bench, curled a right-footed shot into the top corner from outside the area.
Wycombe gave themselves a glimmer of hope when Matt Bloomfield pulled a goal back with seven minutes remaining but the Robins' defence held firm.
Seventeen-year-old Ben Tozer, making his FA Cup debut for Swindon, was prominent amongst the defensive heroes.
Looking to the next round, Sturrock asked for: "Just a home game. The important thing is to play at home."
For Lambert and Wycombe though, the dream is over for this year.