Matias latched on to a goal-kick, slotting in the opener after half an hour, after Alim Ozturk hesitated and failed to clear.
With ten minutes to go, Wednesday secured the game when Barry Bannan re-worked a poor corner to cross wonderfully for Reach to glance a header into the far corner.
The win for the Championship club sets up a second-round tie at home to Premier League new boys Wolves.
Sunderland started the better side and looked the more likely to score the game’s first goal.
Elliot Embleton had the Black Cats’ best chance when he was played through on goal. The linesman’s flag didn’t go up but the 19-year-old dragged his rushed shot wide.
Despite having 12 shots in the game, the home side only managed to get one on target whereas the visitors were far more clinical.
Jos Luhakay’s team scored with their first real opening – Ozturk failed to cope with the bouncing ball over the top and Matias coolly found the net.
Jack Ross’ side had a few further half chances before the break but a lack of quality was evident from the hosts.
In the second period, the Championship side controlled and it was only a matter of time before they sealed the win.
And it came from, ironically, star man Bannan’s worst corner of the night. It came back out to him and he delivered a whipped delivery, begging for a team-mate to get a touch on it and turn it home – and Newcastle fan Reach duly did so.
It was Wednesday’s first win of the season and will be hoping this can kick-start their season, ahead of their trip to Brentford on Saturday.
For Sunderland, focus can now return to this season’s priority: an immediate return to the Championship following.
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