And Rangers, who were reported to have approached the administrators of Crystal Palace this week to ask for permission to speak to Eagles boss Neil Warnock, were unable to mount a recovery as they slipped to a sixth loss in seven games.
Robson opened the scoring from the spot in the 39th minute after Peter Ramage had been penalised for a foul on Jeremie Aliadiere.
And the former Celtic midfielder added a second from 12 yards six minutes later following Kaspars Gorkss' heavy challenge on Leroy Lita.
Rangers hit the woodwork twice through Jay Simpson and Mikele Leigertwood, but they failed to get on the scoresheet for a fourth successive away game.
Strachan made two changes to his Boro side, with Chris Killen returning from a hamstring problem to replace the injured Scott McDonald and Lita coming in for Julio Arca, who dropped to the bench.
Leigertwood returned from suspension to captain Rangers. The midfielder replaced Matt Connolly, who was ruled out by an ankle injury.
Rangers came agonisingly close to taking the lead in the 14th minute as Simpson struck the post with a shot from the edge of the area.
At the other end, Boro saw appeals for a penalty waved away by Andy Woolmer after Damion Stewart appeared to block O'Neil's corner with his arm.
Leigertwood then tested home goalkeeper Danny Coyne with a low drive from 18 yards, but it was the Teessiders who looked the most likely to open the scoring as the opening period drew to a close.
And so it proved six minutes before the break as Aliadiere - who had spurned a glorious opportunity moments earlier - was brought down in the box by Ramage, handing Robson the chance to blast home the resulting penalty.
Simpson saw an effort parried away by Coyne as Rangers looked to respond.
But Boro added a second in the 45th minute, Robson once again making no mistake from the spot after Gorkss had gone into the book for a strong challenge on Lita.
There was little in the way of goalmouth action after the interval as Middlesbrough protected their two-goal advantage with relative comfort.
Stewart did head wide for the visitors in the 64th minute, but they rarely looked like finding a route back into the game.
And Lita almost added a third for Boro, only for Ikeme to palm away the forward's powerful strike.
Rangers' afternoon was summed up with 12 minutes remaining as Leigertwood connected with Ramage's cross but saw his shot bounce back off the post with Coyne well beaten.