League One Carlisle United were second best all afternoon and Swindon thoroughly deserved their place in the second round draw.
Swindon midfielder Curtis Weston had the first shot of the day in the sixth minute. He cut inside from the left-hand wing and hit a right-footed drive that Keiren Westwood managed to hold on to.
Swindon took the lead after nine minutes when Lumsdon scored a diving header – but unfortunately for the Carlisle midfielder it was into his own net.
Aaron Brown put over a teasing cross from the left and Lumsdon dived and headed the ball home from seven yards past goalkeeper Westwood.
On 11 minutes Swindon defenders Jerel Ifil and Jack Smith got into a tangle and almost let Carlisle striker Derek Holmes in. Ifil recovered to hack the ball clear just as the mistake looked like leading to a goal.
Swindon striker Roberts tried his luck from 20 yards in the 14th minute but his right-footed drive flew well over the bar.
Royce Brownlie had two opportunities to stretch Swindon's lead but the Australian striker could not take them.
Ricky Shakes played Brownlie through down the right but his strike was smothered by Westwood in the Carlisle goal with Brown at the far post screaming for the ball when unmarked.
Brownlie's second chance came in the 26th minute but Westwood made a fine save from his 12-yard strike.
Carlisle equalised just before half-time courtesy of captain Kevin Gray's towering header.
Lumsdon made partial amends for his mistake by putting over a fine corner and Gray leapt highest to power the ball into the net. Paul Evans desperately tried to clear the ball, but Swindon's diminutive midfielder could not keep it out on the line.
Swindon came out after the break with the bit between their teeth and harried Carlisle all over the pitch.
Striker Roberts rampaged down the right-hand side beating two defenders before putting over a teasing cross but Brownlie just failed to get on the end of it as Westwood nicked the ball off his toes.
Swindon deservedly took the lead on 51 minutes when Danny Livesey tripped over and let Roberts in. The Robins hitman took the ball on and made no mistake as he drilled it home from just inside the area.
Shakes missed a glorious opportunity to wrap up the win on 85 minutes. A cross came to him unmarked seven yards out but the midfielder put the ball a yard wide of the left-hand post.
Carlisle was struggling to keep up with the quick passing and movement of the Swindon team and it was no surprise when the Robins extended their lead with Roberts' second goal.
Substitute Lucas Jutkiewicz played him through and Roberts danced through a gaping hole in Carlisle's defence before smashing the ball past Westwood and into the bottom corner.
The third goal was well deserved and welcomed Sturrock into the County Ground hot-seat in impressive fashion on an afternoon to savour for the Robins' fans.