The ex-Pools winger, who moved to Brunton Park in 2009, drove in a low, angled finish to sink the 10-man home side with five minutes to go.
Pools, took the lead five minutes before the break when James Poole latched onto Steve Howard's flick to drill home.
The Cumbrians missed a glorious chance to level after 51 minutes when Jake Jervis went through on goal and rounded keeper Scott Flinders, but faced with an empty net he could only put the ball in the stand.
Howard hit the bar after 73 minutes, and two minutes later the scores were level. Robson's free-kick was parried by Flinders and Danny Cadamarteri gobbled up the rebound from close range.
Pools defender Neil Austin was dismissed on 77 minutes, collecting a second caution for a foul on David Symington. And, with the home back four still reorganising, Robson took advantage to fire in the winner.