The poll has a margin of error of plus or minus 3.3 percentage points.
Earlier polls showed Strickland trailing Kasich by double digits.
In the race for Ohio's open U.S. Senate seat, the new poll shows that GOP candidate Rob Portman has a 58-39 lead over Democrat Lee Fisher.
The University of Cincinnati's Institute for Policy Research surveyed 839 likely Ohio voters for the poll between Oct. 14 and Oct. 18.