The visitors took the lead in the third minute when Robinson sent Connor clear and the striker easily beat Lee Worgan from close range.
Swansea's second goal came after 20 minutes when Robinson beat captain Graham Allen to the ball on the edge of the penalty area and then coolly slotted home from eight yards.
Seven minutes later the enigmatic Lee Trundle saw Worgan off his line and tried an audacious 40-yard chip that went narrowly wide.
In the 34th minute defender Izzy Iriekpen rose for a corner only to see his header well saved at the foot of the post by the diving Worgan.
The home side created little excitement for their fans in the opening period and had their first real attack just six minutes before the interval when Craig Dove's 25-yard shot whistled just over the bar.
Diamonds manager Ernie Tippett changed things round after the interval and the hosts made more of a fight of it. There were still very few goalmouth chances but when Rushden created an opening they found Swans keeper Willy Gueret in outstanding form.
On the hour, David Bell picked out Gary Mulligan and his attempted lob seemed to be going goalwards until Gueret managed to claw the ball away.
In the 75th minute Gueret was on hand again when he produced a great save to push substitute Jason Taylor's free header round the post.
In the dying seconds of the game Marcus Kelly saw his mis-hit cross bounce of the top of the bar and away to safety.