With 91 percent of the precincts reporting, LePage, mayor of Waterville, a town of about 15,000, had 194,214 votes, or 38 percent, while Cutler had 186,636 votes, or 37 percent.
"When we won the primary they started calling me the 'big dog.' Well let me tell you something. By morning, we're gonna finish it like Secretariat," LePage told supporters in Waterville early Wednesday, WABI-TV, Bangor, reported.
If either side asks for a recount, state police will collect ballot boxes and vote-tallying teams will confirm the count under the supervision of Maine Secretary of State Matthew Dunlap, who would then certify a winner, the Maine Public Broadcasting Network reported.
Any further disputes would be decided by the state Supreme Court.
In congressional races, Democratic U.S. Rep. Mike Michaud beat Republican challenger Jason Levesque in the state's large 2nd Congressional District. Michaud, 55, winning a fifth term, had 55 percent of the vote compared with Levesque's 45 percent.
Michaud is a member of the Blue Dog Coalition, a group of moderate and conservative Democrats in the House. His district covers nearly 80 percent of the state by area and is the largest congressional district by area east of the Mississippi River.
Democratic Rep. Chellie Pingree defeated GOP candidate Dean Scontras in Maine's 1st Congressional District, exit polls indicated.
Pingree, 55, had 57 percent of the vote compared with Scontras' 43 percent.
The district includes most of the southern part of the state, including Portland and Augusta.