Despite only making 41% of first serves, Murray and Peers held easily throughout the first set and snatched a break to take the set.
But they missed their two break opportunities in the second set while Monroe and Stadler took theirs to level the match before excellent returning from Murray and Peers won them the Champions Tie-break.
Murray, ranked 84 in the world at doubles, is looking bounce back from a disappointing first round exit at the Leon Challenger in Mexico last week when he and Peers were second seeds.
To progress any further at the ATP 250 Us Men’s Clay Court Championship in Houston, he will have to defeat his old partner, Eric Butorac, who together with Jonathan Erlich are the fourth seeds.
Murray is joined in round two at Houston by London-born Inglot and his regular partner, Treat Huey, who are the third seeds.
Inglot and Huey, ranked 39th and 26th respectively, defeated top-40 singles player Fernando Verdasco and his partner Mischa Zverev 6-3, 2-6, 10-5 today.
Both teams served strongly, with both sets going to tie-breaks before the Italians edged the Champions Tie-break to take the match.
Time will tell how if Delgado and Skupski can be more successful playing in new partnerships than they did together.
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