James Beattie had fired the Blades ahead after latching on to Derek Geary's cross on 62 minutes.
But Colchester equalised straight from the restart when Mark Yeates found Kevin McLeod at the far post.
Michael Tonge blasted the home side back in front from a corner with eight minutes left before Platt levelled when he found the top corner from six yards.
Sheffield United manager Bryan Robson:
"Once we got the second, I thought that would be enough. But Matt Kilgallon, who had an outstanding game, just switched off to allow a free header.
"Hopefully we'll learn from that and won't allow it to happen again."
Colchester boss Geraint Williams:
"Every player out there proved that they want to be at Colchester United."
On debutants Teddy Sheringham and Clive Platt, he added: "I didn't think Teddy would last 90 minutes, but his fitness is amazing. Clive's finish was great, it's a dying art what he did."
Sheff Utd: Kenny, Naysmith, Kilgallon, Lucketti, Geary, Montgomery, Hendrie (Tonge 66), Gillespie, Leigertwood (Armstrong 66), Beattie, Sharp (Webber 87).
Subs Not Used: Bromby, Stead. Booked: Leigertwood.
Goals: Beattie 68, Tonge 82.
Colchester: Gerken, Duguid, Baldwin, Elokobi, Connolly, Jackson, Izzet (Guttridge 87), Yeates, McLeod, Sheringham, Platt.
Subs Not Used: Cousins, Granville, White, Wordsworth.
Goals: McLeod 68, Platt 89.
Att: 26,202
Ref: M Jones (Cheshire).
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