Second seeds Dodig and Melo are possible opponents in the next round, although home fans hope to see local favourite Jamie Marray and Frederik Nielsen cause an upset.
Meanwhile in the Ladies Singles, Camila Giorgi bowed out in the third round after she was convincingly beaten 6-2, 6-2 by fifth seed Caroline Wozniacki on Court 1.
An error-strewn performance saw the Italian send down no less than nine double faults, while she was unable to muster a single ace in a match last one hour and 11 minutes.
Wozniacki said: "I was very happy to win this one. She is such a tricky opponent, she hits so hard I was really pleased how I managed to keep my composure in the end."