Newport took an early lead through a Tom Hancock penalty but Sam Davies levelled the scores four minutes later with a penalty of his own.
Whites centre Aled Jenkins showed great pace to sprint through the middle but he wasted an overlap out wide and although Rhys Williams and Rhodri Hughes kept the move alive, the Newport defence regrouped and were able to snuff out the danger.
The Black and Ambers launched the next attack with winger Alan Awcock and scrum-half Rhys Downes both making breaks before Baker crucially intercepted a pass out wide to bring an end to the move.
Hancock kicked his side ahead with a second penalty goal after referee Sean Brickell reversed his decision for an off-the-ball incident before Newport scored the first try of the match on the half hour mark. Hancock beat a couple of tackles, found centre Geraint O'Driscoll in support who in turn fed Awcock for a well taken touch down under the posts. Hancock converted.
The home side dealt Swansea a devastating blow on the stroke of half-time when Leadbeater made a break into the twenty-two before the forwards drove up to the line and lock Adam Frampton stretched over to score a second Newport try. Hancock converted for a half-time lead of 20-3.
Visitors' coach Mefin Davies must have given his side a talking to at half-time as they came out on fire after the interval. The impressive Jenkins burst through a tackle and drew in a defender to send Nicky J Thomas over for a try. Davies converted.
Five minutes later, the Whites were right back in contention when flanker Rob Dudley-Jones powered towards the line from a quickly taken free kick before the ball was swept out wide for Rhys Williams to score in the corner. Davies once again added the conversion.
Hancock was handed an opportunity to extend his side's lead when James Bija held onto the ball after the tackle but his effort at goal fell short of the posts. Rhys Williams pounced on a loose ball inside his own twenty-two and hacked the ball upfield, but his final kick went dead when a try looked certain.
Leadbeater gave his side some breathing space with a try out wide after a long period of pressure. Hancock converted. The Black and Ambers' earned a bonus point when centre Will Richards showed good pace out wide before linking up with prop Jamie Jeune. The pack drove forward and Richards regathered the ball to squeeze over in the corner. O'Driscoll missed the conversion.
Replacement Brendon Lampitt rounded off an outstanding victory for the home side when he took the ball from a lineout and cut through a tired Whites defence to score. O'Driscoll converted.