Izale McLeod clinched Crawley their third successive 1-0 win with a second half goal from the penalty spot.
Reds earned three points from their first League One home game of the season from a tremendous breakaway move in the 51st minute when McLeod was felled by Swindon’s Louis Thompson and referee James Linington pointed to the penalty spot.
He also reached for his red card to reduce Swindon to ten men for the rest of the second half.
Reds’ keeper Brian Jensen earned the Man of the Match award for pulling off a series of saves against a dominant Swindon to keep his side in the game.
‘The Beast’ began a briliant display of goalkeeping with a great save to turn over a powerful shot from the oncoming Louis Thompson from 25 yards.
Ben Gladwin kept the pressure on as Swindon looked for an opening goal as he was denied by Jensen and then Sonny Bradley and lifted another attempt over the bar.
John Gregory made two changes to his starting line-up which beat Ipswich Town 1-0 in the Capital One Cup with striker Gavin Tomlin making his league debut while defender Ryan Dickson returned.
They replaced Bobson Bawling and Lewis Young who were on the bench.
Tomlin played up front with the match-winner from the opening two games, Izale McLeod in a 4-4-2 formation.
Dickson played in left midfield while Gwion was on the right.
Swindon manager Mark Cooper made one change to the team which beat Luton Town 2-1 on Tuesday with Jack Barthram replacing injured left back Brad Smith.
It took until 35 minutes before Crawley had a decent chance when Tomlin had a turn and shot from eight yards turned away by Robins’ keeper Wes Foderingham after a good cross from the right by Edwards.
Moments later Massimo Luongo’s first-time shot was kept out an by alert Jensen.
Crawley wasted a great chance on the break as Dickson’s great cross set-up an unmarked McLeod but his header flew over the bar.
Tomlin was then booked for diving after he seemed to delay going down from a challenge in the area.
Swindon continued to threaten at the start of the second half as Michael Smith fired on target an effort which was deflected over the bar by Bradley.
Luongo blasted another shot towards the Crawley goal from just eight yards but again Jensen was up to the task with a great save.
Crawley took the lead against the run of play on the break when Jensen kicked a long ball out for McLeod to run into the area with.
He was brought down by Louis Thompson and a penalty was awarded, while Thompson was shown a red card. McLeod then scored his third goal of the season with a calmly taken spot-kick struck to the keeper’s left.
Despite being reduced to ten men for the last 39 minutes, Swindon continued to attack and might have equalised if not for poor finishing by Yasir Kasim.
He got no power on an effort which Jensen easily gathered and then put another attempt well wide.
The hard-working Gwion Edwards nearly added a second goal with a double attempt from inside the box - the first was blocked before bouncing back to him and the second hit the crossbar.
REDS: Jensen, Oyebanjo, Dickson, Henderson (Bawling 76), Leacock, Bradley, Edwards (O’Connor 85), Smith, McLeod, Walsh, Tomlin (Banya 62). Unused subs: Young, Rose, Richefond, Spiegel
SWINDON: Foderingham, N Thompson, Byrne, Luongo, Turnbull, Gladwin (Williams 48), Kasim, M.Smith, L.Thompson, Barthram (Randall 67), Rossi-Branco
Unused subs: Lelan, Waldon, Barker, Belford, Randall, Marshall
ATTENDANCE: 2,710 (667away)
MAN OF THE MATCH: Brian Jensen