Mike Jones stabbed home the opening goal of the game on 10 minutes before Wilkinson extended the lead less than a minute later.
Massimo Luongo struck on the hour mark to reduce the arrears but Latics defended resolutely and came away with the victory.
Latics started the game on the front foot and took the game to their opponents from the kick off.
Their pressure paid off on ten minutes as Latics took the lead through Jones. Wilkinson broke the offside trap, raced to the byline and crossed for the onrushing midfielder to confidently place the ball past Wes Foderingham.
One became two just a minute later as Latics took a two-goal lead. Wilkinson nicked the ball off the defender and rolled the ball past the goalkeeper from the edge of the box to give Latics the dream start.
Wilkinson's night came to an abrupt end on 15 minutes as he was taken off with a suspected hamstring injury.
Rhys Murphy entered the game to rapturous applause from the home supporters as he made his home debut for the club.
The visitors started to impose themselves on the game and they started to look dangerous in attack.
After the Robins causing a succession of corners, Latics broke and Murphy capitalised on an error in the Swindon defence and forced a corner.
From that resulting set piece, Latics thought they had made it three after Jonathan Forte headed in a Jones cross but the linesman had his flag up for offside.
The visitors ended the half strongly with Rossi-Branco testing the palms of goalkeeper Jake Kean and then he tipped over Michael Smith's header from point blank range.
Luongo, an Australian international, threaded the ball through to the onrushing Jonathan Obika five minutes before half time but Liam Kelly cleared the threat excellently.
Brian Wilson almost made it 3-0 to Latics 20 seconds into the second half. Murphy held the ball up well, knocked the ball to Winchester and the defender struck from 25 yards but it hit the stanchion behind the post.
The hosts were struggling to cope with the Swindon attack but looked very promising on the break.
With the Robins attacking, Elokobi cleared towards substitute Danny Philliskirk who fed Murphy in the box but his attempted lob was saved by Foderingham.
Swindon did manage to pull a goal back just after the hour mark. Luongo was allowed time and space to shoot and found the bottom corner from 25 yards to reduce the arrears.
Latics did not let that goal affect them and almost restored their two goal advantage with 20 minutes to play. Carl Winchester nicked the ball around the last defender but was thwarted by the Swindon goalkeeper.
The crowd were doing their best to get behind the players but it was the visitors that were enjoying large spells of possession.
Robins substitute John Swift should have levelled the game on 80 minutes but his header ricocheted against his own player and away from danger.
It was backs against the walls stuff for Latics but goalkeeper Kean was assured in goal and punched away and quelled the Robins' attacks.
Two minutes from time, Forte had a perfect opportunity to seal the game for the home side but he struck straight at Foderingham.
Deep into additional time, Latics managed to hold onto the ball and saw out the game professionally to seal the victory which takes them to fifth in the Sky Bet League One table.
Oldham Athletic: Kean, B.Wilson, J.Wilson, Elokobi, Mills, Kelly, Jones, Winchester (Poleon 81'), Mellis (Philliskirk 56'), Forte, Wilkinson (Murphy 15').
Subs not used: Coleman, Kusunga, Brown, Turner.
Swindon Town: Foderingham, Luongo, Stephens, Turnbull, Gladwin, Kasim (Swift 63'), Smith (Hylton 77'), Tofollo, Thompson, Obika (Williams 59'), Rossi-Branco
Subs not used: Barthram, Belford, Rodgers
Goals: Oldham Athletic (Jones 10', Wilkinson 11') 2-1 Swindon Town (Luongo 63')
Attendance: 3,451 (233 visiting supporters)