United had an early chance to add to any Celtic nerves, only for Noel Hunt to drive the ball straight at goalkeeper Artur Boruc from the edge of the penalty box.
However, the home side's defence, missing Lee Wilkie and Mihael Kovacevic who were involved in a training-ground bust-up earlier in the week, was soon under pressure.
Manager Craig Levein had insisted that captain Wilkie was being rested to avoid a yellow card that would mean a three-game ban next season.
But he wasn't the only United defender posted missing as Celtic broke with a four on two advantage and Lukasz Zaluska did superbly to tip Aiden McGeady's effort over the bar following a fine Shunsuke Nakamura slide-rule pass.
It was goalkeeper Zaluska's fingertips to the rescue again, denying Barry Robson from 12 yards.
News conference: Celtic boss Gordon Strachan
United were in danger of being swamped but steadied themselves to threaten on the break.
Lee Naylor tracked back just in time to block a Mark de Vries drive and Danny Swanson poked the ball over the crossbar from 10 yards when found at the back post under pressure.
Paul Hartley sent a long-range drive a few feet over as Celtic continued to press, though United had the last chance of the half, De Vries turning Gary Caldwell inside out following a fine Willo Flood pass but slicing his effort wide from 16 yards.
With Rangers also tied at 0-0 at the break and Celtic pushing for a decisive goal, Nakamura was booked for taking a dive in the penalty box as Celtic pushed forward.
De Vries missed a couple of chances for United
United came close when De Vries again wriggled clear in the box, but his effort was deflected into the side netting, while a fine Swanson drive from 20 yards was turned over the bar by Boruc.
But any nerves began to settle as news started to filter through that Aberdeen had scored at Pittodrie.
And Celtic made sure of the title after 72 minutes when Vennegoor of Hesselink rose in a crowded penalty box to head home a fine Paul Hartley corner from eight yards.
A second goal for Aberdeen against Rangers meant Celtic could see out the remaining minutes - and secure the title - with ease.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.