Bartoli edged his fellow Italians Giuseppe Guerini and Paolo Bettini at the finish, with Noe three seconds back in the rainy conditions.
Noe, who rides with Bartoli and Bettini on the Asics team, moved up from second place overall. He took the pink jersey from Laurent Roux, who won Thursday's 12th stage but faltered on the final climb Friday, up to 650 meters (2,145 feet) above sea level.
The race favourite Alex Zulle, who held the overall lead for much of the event, and one of the leading contenders, Marco Pantani, were part of a three-man breakaway up the final climb, which came 10km (6 miles) before the finish. However, both tumbled twice on the slick roads going downhill and were overtaken.
Bartoli covered the 166km (103 miles) from Carpi to Schio in northern Italy in 3hr 58min 2sec. Pantani recovered to finish only 16 seconds behind him, and Zulle managed to restrict the damage done to his challenge by clawing his way back to finish 22 seconds off the pace.
"We worked hard going uphill and then on the downhill stretch we attacked and it went well," said Noe, who claimed his first victory as a professional in Wednesday's 11th leg.
He holds a six-second advantage over Bartoli in the standings, with Switzerland's Zulle lying in third place, a further 21sec behind the new leader.
The hardest stretch of the Giro begins with today's 14th stage, 165km (102 miles) from Schio to Piancavallo, in the Dolomite mountains.
The 22-stage race ends 7 June in Milan.Reuse content