Gary Mackay-Steven gave United a first-half lead but the home side were pegged back by an Aaron Muirhead penalty. Andrew Robertson reinstated the home side's lead before Partick were reduced to 10 men after Conrad Balatoni picked up his second booking of the game.
Mackay-Steven added his second before Graham added United's fourth as they ended the game playing against nine men following Stuart Bannigan 's dismissal.
United made the best possible start to the game - Mackay-Steven getting the home side off to a flyer when he received a pass from Gauld before firing the ball into the bottom right corner from outside of the area.
Moments later, Nadir Ciftci almost doubled the home's advantage but Scott Fox palmed his long-range strike out for a corner.
United continued to dominate with Mackay-Steven and Gauld full of confidence and firing shots on goal at will.
The away side did manage to carve out an effort on goal as the half drew to a close but Kris Doolan's effort failed to trouble Radoslaw Cierzniak in the Dundee United goal.
Dundee United came out firing after the break with Gauld and Ciftci both going close in the early stages of the second half.
But United were pegged back after 56 minutes when Partick grabbed an unlikely equaliser from the penalty spot. John Souttar was penalised for a handball in the area and Muirhead put the subsequent penalty in the bottom right corner.
United regrouped, forcing a succession of corners as they went in search of their second win in consecutive matches.
They had to wait till the 70th minute to reinstate their lead when Robertson rifled the ball home after picking up a pass from Gauld.
Things went from bad to worse for Partick moments later when Balatoni was dismissed after picking up his second caution and just two minutes later they conceded.
United made the game safe two minutes later when Gauld made it a hat-trick of assists by laying on Mackay-Steven for his second of the game.
But the home side were not finished there and after 84 minutes they made it 4-1. This time Gauld found Graham inside the box who let fly rocketing the ball into the top right-hand corner.
Partick's woes were compounded when Bannigan was shown a straight red card for a foul on Mackay-Steven as they slipped to their fourth consecutive defeat.