Paterson, with 40 caps and 210 international points, was preferred to Glasgow hooker Bulloch, who was widely tipped for the job but had to settle for the vice-captaincy against the Welsh for Williams' first game in charge.
The Kiwi coach told the Scottish Rugby Union website: "Chris and Gordon will captain the side in the next 12 months. We see it as a partnership.
"They have a series of lieutenants on the field around them who will be part of the leadership group.
"Chris is an outstanding player, assured of his position. His character as a man and his conduct as a non-smoker and non-drinker - but still with a great sense of humour - is a great example for the team and for the nation as someone we can look up to."
Paterson said: "I'm delighted and deeply honoured. It's an exciting few weeks ahead and the team approach to leadership where we have a group of players involved, some with a lot of experience, underlines that it's about the team, not individuals.
"During the match the game takes care of itself really. Captaincy will not change the way I approach games."
Paterson has captained school teams, the South at under-19 and under-21 and was vice-captain of the Scotland Under-21 team.
Williams added: "Gordon is his vice-captain. He has captained his country, captained Glasgow and will run the forwards and will set the attitude agenda for the team."
Scotland squad for RBS 6 Nations Championship opener against Wales on February 14, kick-off 4pm:
Backs - M Blair (Edinburgh), C Cusiter (Borders), S Danielli (Bath), A Henderson (Glasgow), B Hinshelwood (Worcester), B Laney (Edinburgh), D Lee (Edinburgh), G Morton (Borders), D Parks (Glasgow), C Paterson (Edinburgh, capt), T Philip (Edinburgh), N Walker (Borders), S Webster (Edinburgh).
Forwards - G Bulloch (Glasgow), S Cross (Edinburgh), B Douglas (Borders), S Grimes (Newcastle), N Hines (Edinburgh), A Hogg (Edinburgh), G Kerr (Leeds), C Mather (Glasgow), S Murray (Edinburgh), R Russell (Saracens), T Smith (Northampton), S Taylor (Edinburgh), J White (Sale