All three shot 3-under 69s after the start was delayed by 1 hour and 39 minutes due to rain and wet conditions. The course was hit with over two inches of rain beginning Thursday night.
Organizers switched to a two-tee start — in which some players start on the front nine and others on the back end — to make up for lost time.
"Conditions were remarkably good with all the rain," said Funk, who won the U.S. Senior Open in 2009.
The "course was soft and you could really go at the pins today," said Allen, who chipped in from 25 feet away for a birdie on the par-4 10th.
John Cook and Sandy Lyle of Scotland were one shot behind after 70s. D.A. Weibring shot 70, while a quartet of players — South Korea's Nam-Sin Park, Japan's Naomichi Ozaki, Olin Browne and Zimbabwe's Denis Watson shot 72s.
The Posco E&C Songdo Championship is being played on the 7,087-yard course at the Jack Nicklaus Golf Club Korea in the South Korean port city of Incheon. Nicklaus designed the course.
The event has a record Champions Tour purse of $3 million. The winner will take home $450,000.