Jerel Ifil's spectacular strike in first-half injury time had given Swindon the advantage at Edgar Street.
Matters were evenly poised through much of the first half, although Anthony McNamee was giving Hereford debutant Richard Jackson a torrid time with some direct running.
The visitors wasted a great chance to take the lead on 25 minutes when Billy Paynter's pass sent Simon Cox away, but the striker dragged his shot across goal and wide.
Hereford were restricted to long-range efforts from the returning Andy Williams and Hudson-Odoi, but had just put together their best spell of the half when Swindon took the lead.
From the Robins' third corner of the period, Michael Timlin let fly with a fierce drive which beat Darren Randolph and came back off the bar.
Ifil drilled back the rebound which flashed into the net off the woodwork.
Hereford looked more purposeful at the start of the second period and Hudson-Odoi fired an early chance into the side-netting.
But Swindon regained their stride and a 20-yard shot by Paynter forced a brilliant diving save from Randolph.
The Bulls keeper foiled Paynter again with another fine stop shortly afterwards before Hereford hit back on 68 minutes.
United were awarded a free-kick 20 yards out and, after Kris Taylor's firm drive had come back off the woodwork, Hudson-Odoi pounced to score for the third time in Hereford's three home matches.
Clint Easton sent a searing drive flashing inches wide as Swindon sought to regain the lead and another excellent Randolph save foiled the hard-working Paynter seven minutes from time.
But in the end, honours were shared as they had been on the Robins' last visit to Edgar Street two seasons ago.