The Scotland forward struck after 47, 65 and 84 minutes to put the Hoops on the cusp of a fourth straight title.
Ronny Deila’s side can now clinch the crown if they beat Dundee at Celtic Park on Friday night and second-placed Aberdeen fail to beat United at Tannadice the following afternoon.
Craig Gordon made a fine save from United’s 19-year-old striker Robbie Muirhead in a goalless first half before former Wolves and Hibernian forward Griffiths struck.
Two minutes after the restart he took a pass from Stefan Johansen before drilling an angled drive past Radoslaw Cierzniak and then beating the United goalkeeper from much closer range in the 65th minute to effectively seal the win.
Referee William Collum then adjudged that United skipper John Rankin had brought down substitute and former United team-mate Gary Mackay-Steven inside the box – although there were some suggestions he made the wrong call – which allowed Griffiths to slam in his third goal (84).
A speculative 20-yard drive from teenager Muirhead, making his first United start since signing from Kilmarnock in February, had drawn a fine fingertip save from Gordon before Stuart Armstrong set up Virgil van Dijk at the other end with a cut-back.
However, the Dutch defender’s drive from 10 yards was half-blocked by United stopper Callum Morris with Cierzniak completing the save.
Celtic skipper Scott Brown picked up a booking for a challenge on home counterpart Rankin before the Hoops’ captain was involved in Griffiths’ opener at the start of the second half.
Brown slipped the ball to Johansen who, in turn, quickly put in the unmarked Griffiths who confidently drilled the ball right-footed past Cierzniak and into the far corner of the net.
Whatever anxieties had been expressed in the away dressing room at the break evaporated and a pinpoint cross from Johansen in the 51st minute found Kris Commons whose close-range header was brilliantly saved by Cierzniak.
The Pole followed up that block with further saves from the former Derby player and Johansen, before Van Dijk came close with a trademark free-kick.
A second goal came soon afterwards when Van Dijk headed against a post and Griffiths bundled the ball over the line before taking the acclaim of the travelling fans behind the goal (65).
Mackay-Steven rattled the crossbar with a drive from the edge of the box and the destination of the points was settled when Collum pointed to the spot after the former United man was pulled down by Rankin on the perimeter of the penalty box.
Griffiths coolly drove in the spot-kick to take his tally for the season to 18 and Celtic a huge step closer to a fourth successive league crown with four games remaining.