The Argentine recovered from a dismal opening set to take the second and force a decider, from which the two produced their best tennis in a finale which Murray finally claimed with a break of serve in game 11 before coming from behind to hold and avoid being dragged into a tie-break for the match.
Murray, who turned 25 today, was presented with a birthday cake on court following the win and will meet Richard Gasquet who beat Paolo Lorenzi 6-1 7-6.
The world number four, a semi-finalist here last year, romped through the first set but trailed 3-1 in the third before rallying for victory in more than two-and-a-half hours.
In a testing wind and with the sun dropping behind the stands of the main stadium at the Foro Italico, Murray looked on course for a simple win when he romped through the first set.
However, Nalbandian, ranked 42 but once as high as third, used the drop shot and angled groundstrokes to great effect to level the match and then go up 3-1 in the decider.
Immediately, Murray broke back and then, at 5-5, took his third break point chance to lead 6-5 before saving two break-back points to reach round three.
"After the last couple of weeks I have had, I am so happy to win," said Murray, who missed Madrid with a back injury.
"Getting the break back immediately in the third was important and I got a bit lucky at the end but I went for it and sometimes you make your luck."