Costa was trailing 3-6, 3-2 when bad light stopped their first round match on Tuesday but he was transformed when play resumed on Wednesday as he ran out a 3-6, 6-3, 6-3 winner.
French Open champion Gaston Gaudio also continued his preparations for Roland Garros with a 6-2, 6-1 defeat of Czech Michal Tabara to reach the quarter-finals.
The world number five and second seed revelled on the clay and made light work of an opponent ranked 91 places below him.
However third seed Joachim Johansson was bundled out by Guillermo Garcia-Lopez 6-2, 6-4 in the second round.
Eighth seed Felciano Lopez did survive, though, ousting fellow Spaniard Albert Montanes 7-6, 4-6, 6-4.
The 51st ranked Costa admitted that he was heading out of the tournament before bad light intervened on Tuesday.
"Yesterday he was playing better than me and it helped a lot to stop," he said. "In the break I could rest and start to think and change things that I needed to do properly.
"Yesterday I was finding the wind difficult, especially on my first serve."
Lucky loser Juan Antonio Marin of Costa Rica continued his impressive progress, reaching the quarter-finals after opponent Agustin Calleri retired with a back injury while trailing 6-4, 3-1.
In the women's draw, second seed Dinara Safina celebrated her 19th birthday in style, winning a delayed first round match against Claudine Schaul 6-2, 6-1.
Third seed Gisele Dulko of Argentina beat Alyona Bondarenko 6-2, 6-3 to reach the quarter-finals where she was joined by fourth seed Li Na of China who beat Nicole Pratt 7-5, 6-1.
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