Joachim, 23, who turned professional for the US Postal team at the start of the season, was in a group of 12 riders that escaped the main pack at Wells, 33 miles from the finish of the 106-mile stage.
Ahead lay Cheddar Gorge, the first big climb of this year's race. The five-mile ascent was lined by spectators who were treated to the rare sight of a world-class field in all-out action on British roads.
Joachim and Norway's Bjornar Vestol went clear of the leaders just before the summit line and headed for Limeburn Hill, the last obstacle before Bristol.
"I could sense that Vestol wasn't too happy in the hills and feared he might have a stronger finishing sprint," Joachim said. "So I took my chance and dropped him on the last climb - I had nothing to lose."
Racing under the Clifton suspension bridge and into the final kilometre at Durdham Down, Joachim was struggling to keep his momentum. He just held off a four-strong chasing group led by the Australian David McKenzie of the Linda McCartney team and Italy's Mirko Puglioli of Amore e Vita.
The big loser of the day was Leon Van Bon, who dropped to 18th overall after two days in the red jersey. However his Rabobank team-mate Marc Wauters moved to second overall. "He should have taken the red jersey after being in the breakaway," Van Bon said. "But he failed, and now it will be difficult to make up the 14 seconds deficit."
Chris Boardman had an unhappy day in the hills. Third overall at the start, he finished almost four minutes down and slumped to 21st overall.
The lead can be expected to change again in today's 106-mile fourth stage, from Bristol to Swansea, which is expected to be the toughest of the race.
In yesterday's stage of the Giro d'Italia, Laurent Jalabert increased his overall lead to four seconds. The Frenchman, who had clung to a 0.02sec lead for the last two days, was second in an intermediate sprint to earn a deduction of four seconds from his aggregate time during the Giro's 11th stage, from Sansepolcro.
The stage victory went to the Italian Ivan Quaranta, who was the Giro leader for a day after winning the opening stage at Agrigento. Yesterday he beat the Italian sprinter Mario Cipollini into second place, with the Czech rider Jan Svorada third in a mass sprint finish.
PRUTOUR Third stage (Winchester to Bristol, 106 miles): 1 B Joachim (Lux) US Postal Service 4:16:36; 2 D McKenzie (Aus) Linda McCartney +14 sec; 3 M Puglioli (It) Amore e Vita +14sec; 4 M Wauters (Bel) Rabobank +14sec; 5 B Vestol (Nor) Acceptcard +25sec; 6 L Lebreton (Fr) Bigmat Auber 93 +1:56; 7 N Larsen (Den) Home-Jack & Jones +2:11; 8 . Jonker (Neth) +2:11; 9 M Andersson (Swe) Acceptcard +2:11; 10 J Voigt (Credit Agricole) 2:11. Leading overall standings:1 Joachim 10:35:13; 2 Wauters +13sec; 3 McKenzie +42sec; 4 Puglioli +59sec; 5 Vestol +1:14; 6 Voigt +1:57; 7 Jonker+2:04; 8 J Vaughters (US) US Postal Service +2:05; 9 R Rumsas (Lith) Mroz +2:12; 10 Andersson +2:27.Reuse content