United were left disappointed as they gave away a two-goal advantage on the opening day and, in the end, their fight back wasn't quite enough to earn them a share of the spoils.
There was an early scare for United as Bonham came under pressure from Norris as he chased a back pass into the area. The keeper recovered quickly to whisk it away.
Swindon came again as Mullin jinked his way down the right flank but his low cross was blocked. The big wide man was there for the return delivery but his header ricocheted off Liddle and was easily saved.
The visitors took confidence from their attack and Brophy skipped a challenge out wide to cut the ball back for Norris to shoot. The forward hit it first time but Bonham went long to make the save.
It was a strong start for the Robins and they went close again when Hussey smashed a 20-yard free kick into the wall. The rebound fell kindly for Dunne and Bonham had to concentrate as a low volley came through a ruck bodies.
An attack down the Swindon left saw Devitt and Grainger cause real problems as they battled their way into the area. Lambe almost benefitted but a last ditch tackle stopped him from making the shot.
Home tails were suddenly up and a deep cross from Miller found Adams on the move. He jumped high to head it and Vigoroux had to skip backwards to tip it over the bar. The corner was taken quickly and it somehow made its way through to Joyce, who went for a drilled shot. Parkes put a boot to the stinger but his touch took it wide of the target.
The Robins responded with a quick break and Mullin was disappointed when his unmarked header went straight at Bonham. It was Carlisle's turn to break and Devitt streaked away from his man on the half way line with Lambe in support. The winger kept it all to himself and was unlucky to see his curler rattle back off the upright only to be hooked away to safety.
The action was end-to-end and another excellent Adams cross was cleared at the back stick as Miller lurked. Four Carlisle corners followed but Swindon held their concentration to keep the blue tide at bay.
The first goal of the game came from a regulation set piece which found Norris completely unmarked as he ran off his man inside the box. The striker didn't need to be asked twice as he slotted his firm header past Bonham from close range.
Carlisle pushed for the equaliser ahead of the break but the away side dealt well with the crosses as they came into their box.
Nicky Adams created the first chance of the second period when he whisked a beauty of a ball into the box. The home forwards were on their heels and Purkiss gratefully volleyed it away.
A difficult task became a mountain to climb on 48 minutes when a slip in midfield from Jones left Goddard with a clear run at goal. His rasping drive was parried by Bonham but Mullin reacted quickly to force the rebound into the back of the net.
Devitt tried to make things happen as he nicked possession inside the box but his low cross was cut out by Vigoroux. Carlisle stayed forward and they were rewarded with a lifeline when Luke Joyce hung back for a pass from Hope. The midfielder picked his spot with a well-placed and well-struck shot.
The hosts were let off the hook when they got in a muddle as Liddle, Parkes and Bonham tried to deal with a bouncing ball. Iandolo had it in the back of the net but the flag was up for offside.
Substitute Richie Bennett almost scored with his first touch of the game but his header from the penalty spot went right at Vigoroux. The same thing happened again moments later as Adams put another high one into the box. The big forward got there but it was once again too close to the keeper.
Half chances were created as United pushed forward but tight Swindon defending frustrated the home side as the visitors closed the door to take the opening day points back down the M6 - with Richie Bennett missing a very good header in time added on as he popped a close range header wide of the target. Nicky Adams also went close as he volleyed from 20 yards only for the keeper to make the save.
Carlisle United - Joyce (55)
Swindon Town - Norris (31), Mullin (48)
Carlisle United - Jones (40)
Swindon Town - Mullin (67), Iandolo (77), H Smith (90)
Carlisle United - Bonham, T Miller (Bennett 65), Liddle, Parkes (Ellis 75), Grainger, Lambe, Joyce, Jones, Adams, Devitt, Hope (S Miller 76). Subs - George, Cosgrove, Salkeld, Egan.
Swindon Town - Vigoroux, Purkiss, Hussey, Lancashire, Mullin, Dunne, Norris (H Smith 31), Goddard, Brophy (Iandolo 60 (Thomas 82)), Linganzi, Conroy. Subs - Henry, Robertson, T Smith, Twine.
Referee - Andy Haines
Attendance - 6,036 (557 away fans)