The pair were the only seeds in action on the opening day of the ATP 250 clay-court event and Maximo Gonzalez sprung an upset to send Carreno-Busta out early on.
The first match of the tournament saw the Spanish number eight seed go 3-0 down en route to losing the opening set.
Carreno-Busta also fell 2-0 behind in the second but recovered to draw level, only for qualifier Gonzalez to finish strongly and clinch a 6-3 4-6 7-5 win.
Fifth seed Thiem was also given a stern test by Joao Souza of Brazil in a rain-delayed encounter.
The Austrian world number 50 moved ahead despite making life hard for himself with three double faults in the opening set.
Souza drew level but Thiem broke his opponent three times in the final set to prevail 7-6 3-6 7-5 and book his place in the second round.
In a battle of two qualifiers, Albert Ramos edged through against Croatian opponent Viktor Galovic 6-7 (5-7) 7-5 6-2 in two hours and 29 minutes.
World number 732 Galovic pushed the Spaniard all the way - having moved ahead with the help of six aces in the first set.
Ramos ultimately won the final five games of the match and is joined in round two by Czech world number 70 Jiri Vesely, who needed an hour and 18 minutes to beat Germany's Jan-Lennard Struff in straight sets.
Just eight places separate Dustin Brown and Pere Riba in the rankings and their contest was closely-fought, with the latter winning through 7-6 (7-6) 7-5.
Brown hit 14 aces and won 36 of his 43 first-service points but it was not enough as Riba secured a meeting with fourth seed Andreas Seppi.
Austrian wildcard Gerald Melzer crashed out at the first hurdle, losing 6-2 7-6 (7-1) to compatriot Andreas Haider-Maurer.