Simon Weaver’s men fought back from a goal down to seemingly snatch three points through substitute Kelvin Langmead’s late strike, but they were pegged back in time-added-on.
After an uncertain opening five minutes on their maiden appearance in English football’s fifth tier, Town were quickly into their stride.
Their first foray forward of note saw Ryan Fallowfield advance from right-back and whip a low ball across the face of goal for Aaron Williams, whose close-range effort was deflected onto the woodwork and behind.
Shortly afterwards, a lovely move down the right saw Williams pull back for Jack Muldoon. He dummied the cross, but the onrushing George Thomson just failed to connect.
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With Town really into their groove and Sutton struggling to contain them, another raid down the right ended with Muldoon firing over the top when well-placed inside the box.
Weaver’s troops remained in the ascendancy and went closer still to opening the scoring as Dominic Knowles planted a 26th-minute header against the near post from Thomson’s right-wing free-kick.
Yet, for all their slick football and attacking threat, Harrogate found themselves a goal down six minutes before the interval.
Liam Kitching felled the tricky Tom Bolarinwa close to the left by-line and from the resulting free-kick, big centre-half Charlie Clough climbed highest to power home Aswad Thomas’ inviting delivery.
The effort was Sutton’s first on target and certainly came against the run of play.
With Town rattled, the opening period ended with the visitors on the front foot and both Josh Taylor and Ross Lafayette fired wide of the mark from the edge of the home area.
The second period began with the Amber and Chocolates causing Town problems with their direct approach, however, just before the hour-mark the hosts were back on terms.
They forced a couple of corners and, when the second was cleared only as far as Josh Falkingham, Harrogate’s skipper found Thomson with a fine first-time pass.
The Town winger came in from the left, drove at the defence and drilled the ball goalwards.
Sutton gloveman Jamie Butler couldn’t hold the strike and Muldoon had the simple task of firing into an unguarded net from just a few yards out.
The quarter of an hour that followed was a fairly even affair, but with the match finely poised, Town seemed to find an extra gear.
They upped the ante in the final 10 minutes and were rewarded with what looked as if it would prove to be a dramatic winner four minutes from the end of normal time.
A left-wing corner eventually broke to substitute Langmead and the veteran centre-half rammed home at the far post despite the efforts of a U’s man on the line.
More drama was to follow, however, and with the visitors pressing in injury time, a ball into the box was apparently handled by a home defender.
The referee didn’t spot an infringement, but his assistant did, and with the spot-kick awarded, Jamie Collins converted from 12 yards to earn his side a share of the spoils.