Murray, who with his gold medal triumph at the Rio Games became the first player to repeat as Olympic singles champion, notched his 19th successive match victory by a 6-3 6-2 margin in what amounted to a nice workout for the 29-year-old Scotsman.
The Wimbledon winner scored the first break of the opening set in the eighth game and served out the set.
After seemingly enjoying the long rallies, Murray had his right shoulder massaged on the changeover before the second set and rattled off four games in a row from 1-1 to turn it into a rout.
His victory set up a third-round match against South African Kevin Anderson with a berth in the quarter-finals at stake.
Top-seeded Murray's victory was in contrast to numerous big battles in the U.S. Open run-up event and a slew of tiebreakers.
Third-seeded Spaniard Rafa Nadal, winner of gold in the Olympic men's doubles, avenged a defeat to Pablo Cuevas in February with a 6-1 7-6(4) win over the Uruguayan.
Fourth seed Milos Raonic of Canada won a heavy artillery battle with John Isner 7-6(5) 7-6(5) for his first win after three career losses to the American.
Raonic said he had just spent 10 days in New York working with John McEnroe.
"John makes fun out of the practice, he challenges you in many ways," Raonic said. "He finds ways to point out things that I tried to bring to the match today."
Rafael Nadal (ESP) reacts against Pablo Cuevas (URU) on day five during the Western and Southern tennis tournament at Linder Family Tennis Center.
Image credit: Eurosport
Seventh seed Jo-Wilfried Tsonga of France enjoyed his own double tiebreaker win over American Reilly Opelka 7-6(5) 7-6(3).
Second-seeded Swiss Stan Wawrinka overcame 18-year-old American wild card Jared Donaldson 2-6 6-3 6-4 in a match carried over from rainy Tuesday and will next face his doubles partner Grigor Dimitrov of Bulgaria.
Tiebreakers decided two three-set encounters.
Borna Coric, 19, of Croatia came back from 2-5 in a climactic decider against Australian Nick Kyrgios by winning five of the last six points to claim the tiebreak 8-6.
Kyrgios showed his displeasure after dropping an opening set tiebreaker by smashing three rackets during the changeover.
"You have to focus on yourself," Coric said when asked by Tennis Channel about the excitable Kyrgios. "I believe I did that unbelievably good today."
Next up for Coric is Nadal.
Steve Johnson, who with one more victory would leapfrog Isner to become the top-ranked American player, prevailed over France's Julien Benneteau 3-6 6-2 7-6(5).
Olympic bronze medallist and fifth seed Kei Nishikori of Japan, like Murray, took an easier route with a 6-3 6-2 win over Russian Mikhail Youzhny.
Results from the Cincinnati Open Men's Singles matches on Wednesday
1-Andy Murray (Britain) beat Juan Monaco (Argentina) 6-3 6-2
6-Tomas Berdych (Czech Republic) beat Marcel Granollers (Spain) 6-3 7-6(4)
9-Gael Monfils (France) beat Marcos Baghdatis (Cyprus) 7-5 6-0
4-Milos Raonic (Canada) beat John Isner (U.S.) 7-6(5) 7-6(5)
7-Jo-Wilfried Tsonga (France) beat Reilly Opelka (U.S.) 7-6(5) 7-6(3)
2-Stanislas Wawrinka (Switzerland) beat Jared Donaldson (U.S.) 2-6 6-3 6-4
8-Dominic Thiem (Austria) beat John Millman (Australia) 7-5 6-1
5-Kei Nishikori (Japan) beat Mikhail Youzhny (Russia) 6-3 6-2
Yuichi Sugita (Japan) beat Nicolas Mahut (France) 6-3 7-5
Kevin Anderson (South Africa) beat 13-Richard Gasquet (France) 6-2 6-4
12-Marin Cilic (Croatia) beat Fernando Verdasco (Spain) 6-3 6-1
Borna Coric (Croatia) beat 14-Nick Kyrgios (Australia) 7-6(2) 4-6 7-6(6)
Steve Johnson (U.S.) beat Julien Benneteau (France) 3-6 6-2 7-6(5)
Bernard Tomic (Australia) beat 11-David Goffin (Belgium) 3-6 7-5 6-4
Grigor Dimitrov (Bulgaria) beat 16-Feliciano Lopez (Spain) 5-7 6-3 7-6(6)
3-Rafa Nadal (Spain) beat Pablo Cuevas (Uruguay) 6-1 7-6(4)
Marcel Granollers (Spain) beat Jiri Vesely (Czech Republic) 6-4 6-7(5) 6-2 J
Juan Monaco (Argentina) beat Ivo Karlovic (Croatia) 6-4 6-4
Yuichi Sugita (Japan) beat Alexander Zverev (Germany) 6-7(4) 6-4 6-2
Nicolas Mahut (France) beat 15-Roberto Bautista (Spain) 7-6(2) 2-6 6-3