Playing the Dye course, Tucker, 53, routed Robert Jan of Northbrook, Ill., 5 and 4, and Vershure, who turned 50 two weeks ago, scored a 4 and 3 victory over Randy Cochran of Deerfield, Ill.
They are among four seniors who advanced to the round-of-16. The others are Steve Heckel and Bob Ralston.
Tucker plays Allen Santos of Lowell, Mass., in the fifth round Friday. Santos eliminated John Hickson of West Brook, Maine, 1-up. In 1999, Tucker beat Hickson in the finals on the General course at PGA National in Palm Beach Gardens.
Vershure was the winner in 1990 and 1993 on the Champion course at PGA National, beating Larry Rentz in the finals both times. Next up Friday is Tim Angis of Saco, Maine, who nipped Larry Emery, 1-up. Emery is a former Tucker assistant at Bellerive Country Club in St. Louis, Mo.
The last of six championships in the PGA of America's 50th annual Winter Tournament Program continues through Monday with the winner earning $4,500, the runnerup $2,800 from the $52,130 purse. The tournaments are co-sponsored by the Greg Norman Collection and Reebok Golf.
'I putted better today than I have been this week,' Tucker said, 'but he was very unlucky. Three times he hit into a divot and had difficult recovery shots.' Tucker won four holes with birdies, Nos. 2, 3, 12 and 14. After Jan. birdied the eighth to pull within a hole of the lead, Tucker put the match away by winning four straight holes starting at No. 9.
Vershure, head professional at Quaker Ridge Golf Club in Scarsdale, N.Y., site of the 1997 Walker Cup matches, said he was 'doing a good job as a match play golfer' by getting ahead early, then concentrating on hitting greens the rest of the way.
For example, he said, 'Today, I won the first and fifth holes with birdies, the second and sixth with pars.' That gave Vershure a 4-up lead. He lost the eighth when Cochran birdied, both bogeye the 11th and Vershure ended the match by winning the 15th with his fourth birdie of the day.
Robert Sedorcek, St. Louis, Mo., d. Dan Tzivanis, Stamford, Conn., 21st hole.
Steve Heckel, Carterville, Ill., d. Mike Martin, New Haven, Conn., 3 and 2.
Sam Olson, Redding, Conn., d. Tom Joyce, Long Island, N.Y., 2 and 1.
Cory Crelan, Edgartown, Mass., d. Ray Rash, Dayton, Ohio, 3 and 1.
Allen Santos, Lowell, Mass., d. John Hickson, West Brook, Maine, 1-up.
Jerry Tucker, Stuart, d. Robert Jan, Northbrook, Ill., 5 and 4.
Jim Smoot, Huntington, N.Y., d. Brad Peck, Madison, Wis., 3 and 2.
Bob Ralston, Little Rock, Ark., d. Andy Shuman, Lake Zurich, Ill., 4 and 2.
Rick Vershure, Armonk, N.Y., d. Randy Cochran, Deerfield, Ill., 4 and 3.
Tim Angis, Saco, Maine, d. Larry Emery, St. Louis, Mo., 1-up.
Tony Kelley, Holyoke, Mass., d. Tom Hewes, Red Wing, Minn., 2 and 1.
Rob Labritz, Fairfield, Conn., d. Mike Baker Jr., Bar Harbor, Maine, 7 and 6.
John Hayes, St. Louis, Mo., d. Jeff Hunter, Columbia, Ill., 20th hole.
Chris Campbell, Vero Beach, d. Joe Cordani, Simsbury, Conn., 3 and 2.
Frank Dully II, Salem, Mass., d. Jerry Impellittiere, New Windsor, N.Y., 19th hole.
Ed Humenik, Palm City, d. Dan Fabian, West Palm Beach, 4 and 3.