Paire held on after being in trouble late in the second set, saving two break points in the 11th game and then trailing 4-2 in the tiebreaker. The Frenchman will play either Matthias Bachinger of Germany or Italy's Filippo Volandri.
Garcia-Lopez broke four times to beat Fognini for the first time on clay and level their head-to-head record at 2-2. The Italian had won both previous meetings on clay, including in the third round of the French Open last year.
The Spaniard will play either Paul-Henri Mathieu of France or Moroccan wild card Mehdi Ziadi.
Igor Andreev of Russia defeated Albert Montanes of Spain 6-3, 6-3 and will meet Denis Istomin of Uzbekistan.