He faced a total of seven break points in his first two service games and while Murray could not force the break, he eventually did at the crucial moment, needing just one opportunity in the 10th game of match to seal the set.
The former world No 1 did face fightback as Cuevas earned four break points of his own in the opening game of the second set but battled hard to maintain parity.
Andy Murray feeling 'competitive' ahead of Shanghai
He broke again for a 4-2 lead and served the match out to take victory in an hour and 25 minutes, another performance to suggest he is close to full fitness after two hip surgeries.
He will now face Marius Copil, whom he beat in Washington last year but sobbed on court after the epic three-set battle that last more than three hours - how this match stacks up to that one could tell us a lot about how much progress he has made.