Renshaw leads the general classification by four seconds from Swift and third-placed Italian Nicola Ruffoni.
Mark Cavendish, who suffered a crash midway through Sunday's first stage, is six seconds behind Renshaw in fifth while defending champion Bradley Wiggins is a further four seconds adrift.
"It was a great stage and a great victory," 31-year-old Renshaw said. "It's always good to win and I've already had some victories in the Tour of Britain, so I am very happy.
"It's great to be in yellow. It's a bonus. We came here to work for Mark Cavendish. He had a really bad crash yesterday and he is just trying to recover at the moment, so he gave me a chance and the guys rode for me."
Six men contested the day's main breakaway but the nature of the stage's closing kilometres, a testing climb over the Great Orme before a fast 6km descent to the finish, meant they were unlikely to hold on.
Their advantage, which had reached nearly four minutes, was eventually wiped out by a small group of chasing riders inside the final 3km.
After a number of late attacks Swift launched his final sprint too early and was pipped to the line by Renshaw, who was assisted brilliantly by team mates Michal Kwiatkowski and Niki Terpstra in the final descent.
Tuesday's stage three is a 179.9km ride from Newton to The Tumble.
(Reporting by Tom Hayward; Editing by Steve Tongue)
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