The January signing from Celtic fired Gordon Strachan's side to back-to-back wins for the first time since September.
Strachan won just one of his first eight games in charge but Boroare now unbeaten in six and handily placed just outside the top six.
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It was far from a vintage performance from the hosts, who had to survive a nervy final few minutes having failed to add to their lead, but the manager will not mind after pocketing another valuable three points.
Peterborough, by contrast, are heading in the opposite direction. They remain winless on the road, rock bottom of the table and 12 points from safety.
Scotland midfielder Robson was handed the captain's armband in the absence of the injured Gary O'Neil and made his mark after only seven minutes.
Jonathan Franks swept in a cross from the left and Robson arrived at the edge of the area to coolly slot the ball past Joe Lewis in the Peterborough goal.
Posh would have fallen further behind in the first half but for England Under-21 keeper Lewis, who pulled off fine saves to thwart long-range efforts from Franks and Willo Flood.
Scott McDonald could also have doubled Boro's advantage when he met another cross from Franks but he put his header wide.
But the hosts were given a fright on the stroke of half-time when Lee Frecklington's well-struck shot forced a fine save from Danny Coyne.
Posh continued to improve after the break but, as has been the story throughout the season, they struggled to turn possession into chances.
At the other end the lively Franks managed to fashion a shooting opportunity but his effort flew narrowly wide.
Robson then found Franks in the area with a wonderful cross-field pass but Posh defender Craig Morgan was able to get his tackle in before the winger shaped to shoot.
Robson nearly bagged his second goal of the afternoon when he intercepted a loose ball and charged forward but his curled effort floated the wrong side of the post.
Posh boss Jim Gannon threw on striker Craig Mackail-Smith for the final 10 minutes as his side pushed for an equaliser.
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