Nant were ahead 11-14 until the final minute when Davies struck to break Nant hearts and set-up a second round tie against Usk who edged out Rhiwbina 17-14.
Four east's Trinant, who led Cilfynydd 28-17 with 12 minutes to go, were denied victory in spectacular fashion by a penalty try awarded two minutes from time.
Four south east's Cilfynydd will now progress to play Denbigh in round two.
Outside half Richard Powell bagged an impressive 33-point haul as Nantyglo registered a 73-0 whitewash against visiting Bynea.
Powell crossed twice and slotted ten conversions and a penalty to clinch a second round clash at Ystradgynlais.
Centre Nolan Nicklin bagged a staggering five tries for three east's Brynmawr as they hammered Pontyates 87-0. Brynmawr will play Tumble in round two.
Llanishen full back Grant Morris scored 25-points to help the two east side dispatch visiting Llandybie 80-7 and set-up an interesting second round clash with Newtown.
Nicky Strange scored a massive 31 points including a hat-trick as Blaina defeated Bridgend Sports 61-7 to set-up a trip to Pwllheli in round two.
Centre Gareth Jones bagged a hat-trick for Cardiff HSOB to help them to a 57-25 victory over Ynysddu.
The Old Boys will host big city rivals Fairwater in the second round who edged out Hirwaun 12-10 with a try for youth product Jack Lee.
Number eight Chris Bryan scored a brace as three south east's Nelson claimed a fine 38-8 win over two east's Croesyceiliog and a second round tie with Trefil.
Pencoed travel to Aberavon Quins in round two thanks to a 61-0 defeat of visiting Cardigan and St Peters take on Porth Harlequins after dispatching Porthcawl 40-3.
Tonmawr grabbed 16 tries in creating a new club record 106-12 scoreline against Pontllanfraith as full-back Gareth Morris led the way with five tries, and James Donovan claimed a try brace.
Other tries came from Steve Kenny, Rob Hill, Darren Hawkes, Ross Pugh, Mike Ralph, Adrian Phillips, Ricky Price, Lloyd Thomas and Austin Howells, who also notched 13 conversions as Steve Reynolds and Alan Barlow crossed for Pontllanfraith's tries and Chris Moses landed a conversion.
Tumble also enjoyed a 72-0 win over Division Five (West) basement battlers St David's as winger Nicky Allington led the way with a try hat-trick and Rhodri Jones (2), Phil Griffiths, Richard Rees, Huw Evans, James Plimmer, Jeremy Davies, Mark Kinnard and Dominic Brettle got on the score sheet alongside Steve Hewitt, who slotted six conversions.
Seven Sisters led 24-13 at half-time against Briton Ferry and finished up victors by 39-27. Pembroke beat Gowerton 60-7.
Crymych had a narrower win as they overcame Cefn Cribbwr 14-8 thanks to tries by Robert Davies and Gary Nicholas, converted by Gareth Davies.
Mold survived a last-minute penalty attempt by Monmouth outside half Dave Croudace to move through to the second round.
Croudace had earlier collected 18 points from four penalties and three conversions. But his miss enabled Mold to hold on for a 34-33 victory.
The cup draw saw two local derby ties, with Caernarfon winning at Welshpool and Colwyn Bay defeating Ruthin at home.
The Bay overcame Ruthin 15-13 with tries from centre Dave Hocknell and No 8 Chris Browne. Ruthin crossed for a try by scrum half Iwan Roberts, which was converted by outside half Huw Roberts.
Welshpool and Caernarfon were level at five all at the break, but the visitors ran out comfortable victors by 60-12.
Newtown defeated visitors Deri 39-8. Llandudno ran in nine tries as they crushed Abercarn 55-5. There were first round home defeats for Bala (27-12 to Penclawdd) and Bethesda (36-12 at the hands of Cwmbran), while Cambrian Welfare crushed Bangor 106-0.
Pwllheli were the North's leading scorers as they defeated Talywain 75-15.