How do space rockets maintain stability?

It seems to me that a space rocket is a highly unstable proposition because the centre of gravity is well above the thrust. Launching such a craft ought to be as tricky as accelerating flat out in a muscle car on ice with no traction Control. Granted there are aircraft with inherently unstable dynamics (such as the vtol Harrier jump jet etc) but these have only become possible since the advent of sophisticated computer control which keeps them in the air. The Apollo missions, by contrast, famously got to the moon with less computer power than a digital watch. Could they really pull off this trick in 1969? or is there an inherent stabilising effect applicable to rockets that I've missed?