After two goalless opening games, the Magpies were soon on target, netting twice in the first 10 minutes.
But in a tight match spoiled by a very strong wind they were unable to build on their early advantage as the determined hosts found a way back into the contest.
Both Chorley goals came from well-flighted Adam Blakeman corners.
Courtney Meppen-Walter got up well at the back post to head home the first of these on six minutes.
And he was involved again four minutes later, meeting Blakeman’s pin-point delivery with a firm header which rebounded from the crossbar, Chris Holroyd reacting promptly to tuck the loose ball into the back of the net from close range.
However, as the high-tempo start to proceedings was maintained, the Magpies’ two-goals advantage was halved on 14 minutes.
The wind held up a long punt from keeper Jamie Butler and the resulting pinball in the box fell kindly for Aaron Jarvis, who had time and room to shoot low across Joe Anyon from six yards.
Clever footwork by Holroyd on the edge of the box set up a shooting chance for Josh O’Keefe but his drive from distance flew just too high.
Anyon pulled off a fine save low to his left to deny Omar Bugiel as Sutton built momentum and Chorley did well to hold on to their lead until half-time.
With the wind at their backs, Sutton were always likely to pile forward in the second half and after a decent spell of pressure an equaliser was forthcoming in the 54th minute.
O‘Keefe conceded a free-kick 30 yards out in a central position and United’s set-piece master Robert Milsom clipped his shot over the defensive wall beyond the dive of Anyon and just inside the post.
Both sides had half-chances to nick all three points in the final quarter but Chorley came closest when Ntumba Massanka’s stinging goal-bound shot was blocked by a covering defender.
Magpies manager Jamie Vermiglio was pleased with a point but admitted to a degree of frustration at not having claimed three.
“The boys were brilliant in the first 20 minutes, so it’s a bit disappointing not to have won the game after going two up,” he said.
“But we have more than held our own now against three tough teams and the message from today is that Chorley can compete successfully at this higher level.”
CHORLEY: Anyon, Challoner, Blakeman, Teague, Meppen-Walters, Baines, A Newby (Cottrell 78), O’Keefe, Carver, Holroyd (Massanka 57), E Newby (Dodds 68). Subs (not used): Ross, Hooper.