The world No 4 from Scotland overcame Marco Chiudinelli, the third best Swiss player, by 6-3, 6-4, taking him to within one win of a possible rematch with Novak Djokovic, who defeated Sergiy Stakhovsky 7-6, 6-3.
Last month Murray was close to halting the world No 1 in the semi-finals of the Australian Open. Now in his first tournament since then, he suggested he may be re-building momentum.
"I played well throughout today," he said. "I played a little bit better, so that's progress. Now I just need to keep that going."
Murray established an early break of serve against Chiudinelli to reach 3-1 and consolidated the lead well, serving well and moving better than he had the day before against Michael Berrer.
Then the three-time Grand Slam finalist had complained of feeling unwell, and had gone 3-4 down in the second set against the German. But this time against an opponent ranked outside the top 100, Murray was never in trouble.
He almost broke serve again in the first game of the second set, and earned another break point in the seventh game, only for Chiudinelli to play those points well.
But in the ninth game Murray produced some of his most cogent and controlled rallying, outmanoeuvring his oppoent to reach love-40 and then converting the second break point by luring Chiudinelli into over-hitting.
Murray then closed the match out by winning his service game to 15 and told the crowd how training in Florida had helped him cope with the rising Dubai temperatures, in the nineties on court.
"Conditions in Miami are hotter than this," he said. "I find the conditions here almost perfect. I'd been practising serve there and I think that went well today."
Djokovic was again below his best but managed to beat Stakhovsky for his ninth straight win in 2012.
The top-ranked Djokovic looked rusty at times in his opening win against Cedrik-Marcel Stebe, and the victory over Stakhovsky was anything but routine.
Stakhovsky was 4-3 up in the first-set tiebreaker but two costly unforced errors let Djokovic off the hook and the Serb took the set when the 74th-ranked Ukrainian hit a forehand long.
Djokovic had to come from a break down to win the second set and join Murray in the quarter-finals.
More to follow