The third-seeded Romanian needed just 57 minutes to beat the 72nd-ranked Diyas, who was playing at Wimbledon for the first time.
Halep is the second Romanian woman to reach the quarterfinals at the All England Club, joining Virginia Ruzici, who made it to the final eight in 1978 and 1981. Ruzici is now Halep's manager.
Halep, who has reached the quarterfinals at all three Grand Slams this year, is the only player among the top four seeds left in the draw. No. 1 Serena Williams, No. 2 Li Na and No. 4 Agnieszka Radwanska are all out.
The 19th-seeded Lisicki took an injury timeout while facing break point at 1-1 in the second set and was treated by a trainer on her right shoulder.
After returning to the court, Lisicki saved the break point and held serve, then broke in the next game for a 3-1 lead. She was broken by the 72nd-ranked Shvedova while serving for the match at 5-3, but broke again in the next game to close it out. An emotional Lisicki celebrated by falling to her knees at the baseline and resting her forehead on the grass.
Both players piled up more unforced errors than winners. Lisicki had 33 errors and 29 winners, while Shvedova had 34 and 18.
Former champion Maria Sharapova was also playing for spot in the quarterfinals in the upper half of the draw, facing Angelique Kerber of Germany on Centre Court.