Connolly, 19, doubled Brighton's lead before the interval and completed a dream first Premier League start with an exquisite second in the 65th minute that sealed his side's first win over Tottenham since 1983.
Another terrible error by goalkeeper Hugo Lloris led to Tottenham conceding inside three minutes, the goal compounded by the France captain sustaining a serious elbow injury and requiring oxygen as he was carried off the field.
"I am delighted for the players, we have worked really hard and haven't got the results we could have got but that is life in the Premier League, the margins are small". But as a group it's fallen apart and I think we're at the point where something has to change'. To concede and lose your captain is a massive impact emotionally and for the team.
The France goalkeeper also made an error in Tottenham's last league game, a 2-1 win over Southampton, when he was caught in possession near his own goal line. "I didn't really know how to celebrate - I am speechless", Connolly said afterwards.
"After 30 minutes it was 2-0 and we tried to find a solution but it was impossible".
"He's been knocking on the door", Potter told BT Sport when asked about Connolly.
Spurs will go into the contest as favourites considering that they have a more established side but the Seagulls could be very competitive especially considering that they will be playing on home turf.
Lloris spilled a Pascal Gross cross at the feet of Neal Maupay who tapped in to give Brighton the lead at the Amex after just three minutes. Meanwhile, there was an alarming lack of intensity from a Spurs side that just don't look happy at the moment, and the central defensive partnership of Jan Vertonghen and Toby Alderweireld is looking a real weak spot.
Paulo Gazzaniga was substituted on as Lloris' replacement once the France worldwide was taken from the pitch following several minutes of treatment.
Spurs have been struggling to find consistent form in recent weeks and they now find themselves down in sixth place in the Premier League table.
The away side struggled to gain any momentum and it took until late in the first half for Erik Lamela to have their first meaningful opportunity of the match.