The 20-year-old Mexican-American, who grew up playing on rough public courts in Los Angeles, is playing in his first ATP Masters 1000 series event, having qualified with a straight-sets win over Argentine Renzo Olivo on Tuesday.
Escobedo, ranked 108th, beat the 43rd ranked Evans 7-5, 0-6, 6-3 at Crandon Park thanks mainly to his strong first serve, which included 10 aces.
"This day is very special for me. I qualified for my first Masters 1000 here in Miami and I played a tricky opponent who serves pretty well. It is pretty big to get my first win here," Escobedo said.
Escobedo will now face 25th-seed Fernando Verdasco in the second round and he can expect plenty of support from a strongly Hispanic South Florida crowd.
For Evans, it was another setback after he went out in the second round at Indian Wells.
"He played pretty well. I needed to get on top early in the third set and I think he would have gone away but I didn't," said Evans.
There was more US success against British opposition when qualifier Jared Donaldson outlasted 45th-ranked Kyle Edmund 2-6, 7-6 (7/4), 6-2.
Edmund served for the match at 5-3 in the second set and then had three match points in the following game.
Japan's Yoshihito Nishioka triumped in a two-hour battle with Australia's Jordan Thompson.
Nishioka, who reached the fourth round at Indian Wells last week, saved 11 of 14 break points to secure the 1-6, 6-4, 6-3 victory.
There was an unusual stoppage during German veteran Tommy Haas's narrow 6-7 (5/7), 6-3, 7-5 loss to Czech Jiri Vesely when the game was halted due to an iguana on top of the scoreboard. The creature was eventually taken away by officials, but not before it scampered across the court.
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