The Democratic nominee led 47-41 in the poll including leaning voters taken in the three days immediately following the Democratic convention in Philadelphia.
The new CBS survey showed Clinton with a 46-39 lead when respondents who were leaning in favor of one of the candidates weren't included.
When third-party candidates were included, Clinton polled at 43 percent, Trump at 38 percent, and Libertarian Gary Johnson at 10 percent, when including leaning voters.
Trump had a single-point lead, 44-43, in the CBS poll released last week after the Republican convention in Cleveland.
A Morning Consult poll released Sunday also showed Clinton ahead of Trump following the Democratic convention, 43-40. The Morning Consult poll reversed Trump's four-point lead following the Republican convention.
The CBS poll surveyed 1,131 registered voters by telephone nationally from July 29-31, with a margin of error of plus or minus three points.