Spotted this in the Guardian, not sure what provoked it, but never mind. The claim is that a paper published in 1972 made a remarkably accurate prediction of 0.6C, and this proves we've understood the climate system really well for a long time.
For starters, it should be pretty obvious that if people published enough random papers, some would (with hindsight) turn out to be close to correct. That in itself would hardly prove prescience, although it might be hard to refute a claim by the lucky one. See also investment analysts who claim to be able to "beat the market"...
But I'm not going to go looking for all the silly forecasts that were wide of the mark, which there surely were - people are still churning them out, remember Lovelock's few remaining breeding pairs of humans, or Bryden's AMOC shutdown, or Keenlyside's cooling? - but merely evaluate the Sawyer paper on its merits. Found courtesy of wmconnolley.org.uk the paper doesn't really seem to have much original research, but repackages other work in what looks more like a commentary. He uses Manabe and Wetherald's climate sensitivity estimate of 2.4C, and a predicted increase in CO2 of 25%, to get a warming of 0.25*2.4 = 0.6C by the end of the (20th) century.
While the final number ended up pretty close, there are a number of assumptions/approximations/errors (take your pick) in that calculation. Firstly, there are other forcings! The IPCC AR5 lists other factors which in total magnitude exceed the CO2 effect, though the positives and negatives broadly cancel. But Sawyer doesn't consider them at all. Secondly, the logarithmic effect of CO2 means that a 25% increase should equate to a 32% of the effect of a doubling, which would work out at 0.8C...not a huge difference in forecast, but a big difference in level of understanding! Lastly (perhaps) there is also the small issue of equilibrium versus transient response - the thermal inertia of the ocean means there's a chunk more warming in the pipeline, probably about a third as much again. All these values have substantial uncertainty even now, of course - and although I'd say the the 2.4C sensitivity value still looks pretty good, others disagree and at best it was a lucky guess to get it right back then.
So all in all, it looks like he made a number of significant errors which end up cancelling out, thus resulting in a forecast that hit the bullseye much more closely than can have been reasonably expected.