Both eased through their first sets, then had to fight out tie-breaks in the second before each won 6-1, 7-6 against Kent Kinnear and Danny Sapsford respectively.
Bates took advantage of Sapsford's wavering serve in the opening phase, but the British No 1 twice had to battle to hold his serve in the second set against his Davis Cup colleague - and trailed 5-2 in the tie-break before winning 12-10 on his fourth match- points.
'I was more pleased with my form at the start of the match than I was at the end,' said the 31-year-old Bates, who last month became the first Briton to capture an ATP Tour title since 1977 when he won the Korean Open in Seoul. Bates meets the 12th seed, Paul Kilderry of Australia, in the third round.
Edberg found himself 3-0 down in the second set to Kinnear, the 6ft 5in American, and was 0-40 down on his serve in the 10th game before taking the tie-break 7-4.
There were times in the second set when the twice Wimbledon champion looked frustrated with his on-court fortunes at the end of a forgettable month, in which he was dumped out of the French Open in the first round.
'You can't keep up winning for 12 months a year. Your form must dip at some stage and May has been a bad month for me. You could call it a bit of a disaster,' Edberg said.
'But you've got to forget about the whole thing, dig deep and get it out of your system, or it can stay on your mind. It depends on the individual.'
Edberg, 28, will now face Paul Hand, from Berkshire. He pulled off one of the shocks of the day by putting out America's 15th seed, Todd Nelson, some 378 places above him the world rankings.Reuse content