Harrison was thwarted by the fingertips of the United custodian on 85 minutes as we chased a leveller, following Ashley Fletcher’s 58th minute breakthrough for the hosts.
An absorbing game saw chances at both ends, but ultimately our lads were unable to make any of them count.
Harrison went close in the first half following a cross from the right, while the lively Anthony Georgiou also nearly broke the deadlock.
We enjoyed much of the possession while the United forwards found centre-half Cameron Carter-Vickers in great form.
The best chance of the first period went to United on the stroke of half-time as Demitri Mitchell went clean through behind the defence on the counter-attack, but goalkeeper Harry Voss made himself big and kept the ball out.
The home side did, however, take the lead on 58 minutes when another quick break led to Fletcher converting from six yards out after good work in the build-up by Mitchell and Joe Riley.
We continued to look threatening despite the setback, substitute Daniel Akindayini going close, while the home side might have added a second had Kyle Walker-Peters not cleared Mitchell’s effort off the line.
With five minutes left, Harrison worked himself into a shooting position and let fly – but a fine fingertip save from keeper Pereira meant we ultimately left empty-handed.
Spurs: Voss, Campbell-Young (Akindayini 62), Amos (Ross 46), Walkes, Maghoma, Carter-Vickers, Walker-Peters, Goddard (Hayford 78), Harrison, Miller, Georgiou. Substitute (not used): Priestley (GK).