McGregor set the Dolphins' all-time record for passing touchdowns with 41, was 13 of 24 with 294 yards.
Jacksonville (3-3, 2-1 Pioneer Football League) also had a record-setting day on the ground as Rudell Small ran for 52 yards on 15 caries to become the Dolphins all-time leader rusher with 2,379 yards.
Josh Philpart, who opened the scoring on a 29-yard reception from McGregor, finished with 192 yards and two touchdowns on six catches.
The Dolphins took a 14-0 lead as Jason Wood had Jacksonville's first turnover of the season, returning an interception for a 52-yard touchdown.
Morehead State (2-5, 0-4) finished with 140 yards of total offense and had three quarterbacks combine for 88 yards passing and six sacks.