Counties, who had the greater share of possession in the opening half, went on to outscore the visitors by three tries to one.
Their first came after Scottish referee Graeme Wells adjudged that Ireland had kicked the ball out of a five-metre scrum.
Wigglesworth, who had previously been ruled offside from a Chris Johnson cross-field kick, got the try he had been threatening just before the break.
But Clontarf out-half David Joyce made sure to bring his kicking boots with him to bitterly cold Darlington - his third successful penalty made it a three-point game in the 49th minute.
Even better followed as Lansdowne's Mark Roche brilliantly tore through the home defence to round off a fine counter attack. Joyce was unable to convert the centre's try, but Tom Tierney's men had nipped ahead at 14-12.
Centre Jack Adams' converted try under the posts saw the lead change hands again, as he followed up on a break by replacement Gavin Jones.
But two penalties in a three-minute period from the reliable right boot of Joyce - the second a superb kick from upwards of 50 metres - edged the visitors in front again.
The Irish number 10 then dropped a long range penalty short with 10 minutes remaining, as this closely-fought game went right down to the wire.
Luke Myring swung it back in Counties' favour at 22-20 with three minutes to go and just when it looked like the Irish challenge was over, their persistence and sheer will to win paid off with a last-gasp penalty.
The industrious Irish pack forced a turnover and England's discipline let them down when it mattered most, with referee Wells continually picking up on penalties at the breakdown.
Joyce coolly stepped up to slot the match-winning kick from 30 metres out, completing a very successful two-match series for the Ulster Bank League's leading lights.
Joyce's Clontarf club-mate Michael McGrath also deserves credit for his strong running throughout, particularly during the closing stages as the powerful winger clawed back some crucial territory.
It was a bitterly disappointing finish for the Counties players who gave their all in what was a cracking contest. Unfortunately for them, they were left to reflect on a second successive one-point defeat - they were also pipped by a France Federale team last month.
The Ireland Club side had beaten England Counties three times before at home, but this was the first time to taste victory on English soil.
It was something that Tierney's talented squad had targeted this season, especially after they swept past Scotland (28-3) so convincingly in Cork.
They are the second Ireland Club team to complete a clean sweep of victories over their English and Scottish counterparts, following on from David O'Mahony's side back in 2011.
Referee: Graeme Wells (Scotland)
Ireland Club XV: Daniel Riordan (Old Belvedere RFC); Michael McGrath (Clontarf FC), Stuart Morrow (Ballynahinch RFC), Mark Roche (Lansdowne FC), Cian Aherne (Lansdowne FC); David Joyce (Clontarf FC), Gerry Hurley (Cork Constitution FC) (capt); Colm McMahon (St. Mary's College RFC), Tyrone Moran (Lansdowne FC), Declan Lavery (Old Belvedere RFC), Dean Moore (Old Belvedere RFC), Fergal Walsh (Terenure College RFC), Willie Earle (Lansdowne FC), Charlie Butterworth (Lansdowne FC), William Ryan (Cork Constitution FC). Replacements used: Ian Hirst (Clontarf FC) for McMahon, Andrew O'Driscoll (Cork Constitution FC) for Moran (both 51 mins), Barry Daly (UCD RFC) for Morrow, Sean Walsh (Cashel RFC) for Moore (both 62), Neil Cronin (Garryowen FC) for Hurley (78). Not used: Hugh McGrath (Young Munster RFC), James Ryan (Cork Constitution FC), Ritchie McMaster (Queen's University RFC).