The visitors started brightly and had a shot cleared off the line and another effort hit the post, but it was Traf who took the lead through a Liam Salt free-kick on the half-hour mark.
Ten minutes later Pier drew level when Sam Hyde rose highest to head home a Martin Cooper corner, the 1-1 half-time scoreline a fair reflection of an even opening 45 minutes.
The away side started the second half strongly, but were reduced to 10 men when Cooper was shown the red card by ref Ben Naylor for foul and abusive language.
Traf forward Jimmy Beadle then made it 2-1 to Traf with a fine finish, but the hosts were then reduced to 10 men when Alex Sheader was dismissed after an incident involving Pier striker Luke Delve.
The latter then equalised for Pier and looked to have earned them a point, but the visitors still missed two late chances and were made to pay late on when Exley latched onto a clearance by his keeper and raced clear to slot in the winner.
Traf boss Col Jenkinson said: "Every one of my team played very well against a very strong Pier side today and I think we deserved to win, Josh Young and Jimmy Beadle looked particularly good for us."
Pier manager Jamie Towse said: "I am very proud of all 14 of my lads today, they all played their hearts out and we at the very least deserved a point.
"Left-back Tommy Barker was in outstanding form for us today, and Ryan Baldry, in a defensive central midfield role, also had a cracking match."
Newlands maintained their 100% record with a 10-0 win against Cayton Corinthians.
Danny Glendinning scored four for the title-chasers, while central midfielder Liam Cooper scored the perfect hat-trick, netting a header and shots with his left and right foot.
Drew McCoubrey also struck a brace with Ben Luntley's goal completing the rout.
Cooper and his fellow central midfielder Jack Ramos were singled out by boss Joe Hakings as Newlands' star men, with sub Zak Hansen also impressing for the victors.
Roscoes Bar edged away from the relegation zone with a 2-1 home win against Fylingdales at a windy Oliver's Mount.
The hosts, who were playing in a new 4-3-3 formation with Gary Hepples, Dan Thomas and Mark Kenyon up front opened the scoring when Kenyon back-heeled a through-ball to Hepples, who turned his man and fired a left-footed strike into the bottom corner.
The villagers levelled five minutes before half-time when Mikey Kilpatrick slotted in at the far post.
Only 10 seconds into the second period Thomas rounded Dale keeper Jamie Wassall but hit the post, and shortly afterwards Jack Worthy ran all the way from centre-back after winning the ball, only denied a superb solo goal by a brave save from the impressive Wassall.
With 15 minutes remaining, midfielder Matty Rowley sent a through-ball to Thomas, who beat his man and scored with the outside of his left foot.
The Roscoes man of the match was Sean Rowley, who won everything in midfield, while forwards Hepples and Thomas also played well.
Wassall had a good game between the sticks for Fylingdales and kept the visitors in the game, especially during the second half.
Crown Tavern continued their improvement with a 3-0 home Division Two win against Scalby.
Steve Molloy opened the scoring for the hosts in the first half, and after the break Tavern dominated with Dave Barber making it 2-0 after latching onto a poor pass from the Scalby keeper.
Ten minutes from time a free-kick by Tavern's man of the match Chrissy Hannam was headed in by James Burrows to seal the win.
Ayton Reserves came back from 2-0 and 3-1 down to earn a 3-3 draw at Fylingdales Reserves.
The hosts went 2-0 up early on, Lewis Peel scoring the opener from a cross, the second coming courtesy of a Sam Wiggins free-kick.
As the first half went on Ayton fought back and a Jamie Heritage toe-poke pulled it back to 2-1, but Peel struck again with a long-range shot to restore fylingdales' two-goal lead before half-time.
Ayton started the second half brightly and most of the half was played in the Fylingdales half.
The visitors created numerous chances, and soon gained their reward when Richard Howes netted a screamer from 20 yards.
Then a free-kick from Dave Gaddass set Si Coupland up for the equaliser and Ayton were unlucky not to go on and get all three points, only a couple of fine saves from Fylingdales keeper Neil Purves denying them a winner.
Boss Steven Frederiksen said: "This was a fantastic effort from us as we were by far the better team in second half.
"Nicko Dunn was man of the match with a good performance from Danny Milne and Heritage in the middle of park."
Will Stuart-Cross was named as the Fylingdales man of the match.