Swansea may be struggling in the league but recorded an impressive win over Dundee last weekend and will be in confident mood for tonight's game.
Despite the return of their Sevens trio, Carmarthen are struggling with a number of injuries though Williams is confident his side can get their cup campaign back on track with a win.
"We performed well last week considering the strength of the opposition and the injuries to our squad," he said.
"It was disappointing not to get a win but we have a chance to keep our campaign alive with a local derby tomorrow night.
"Sunday's match was our first competitive cross-border game and it was a great experience for our boys but everyone loves a local derby and playing Swansea will be a lot more familiar for us. We have a young side, as do Swansea whereas Cornish Pirates were more mature and experienced.
"It would be great to see some extra support for us tomorrow night in the first of two matches against Swansea, first in the British & Irish Cup then away in the Premiership."
Former Wales lock Williams is relatively happy with his side's progress so far this season but believes there is more to come.
"We haven't started this season as well as we finished last season but we have had some good wins and we can only learn from the little mistakes we've made," he added.
"We're not quite there yet but we are giving young boys an opportunity and they have taken their chances well."