But the game ended with a heated flash-point as United’s Gaetano Berardi and City’s Matty Taylor were both shown straight-red cards after clashing with 11 minutes to go.
Saiz fired United into a fourth-minute lead after good work from Kalvin Phillips and Lasogga and then doubled his tally just ten minutes later after Gjanni Alioski caused a scramble in the City defence from Eunan O’Kane’s through ball.
And after Saiz had missed a near enough open goal for his hat-trick, Leeds bagged their third in the 67th minute when Lasogga clinically converted a Kalvin Phillips corner.
The impressive victory ended a run of three-straight defeats in emphatic style as Leeds moved back up to fourth.
United raced into a fifth-minute lead through Saiz after Phillips had charged down a weak clearance.
The ball fell to Lasogga who played in Saiz whose shot took a deflection to beat ‘keeper Frank Fielding en route to the bottom corner.
Leeds were dominating and Saiz then saw his tenth-minute cutback from the byline deflected narrowly wide for a corner which City survived.
But sloppy City defending then led to a second United goal in the 15th minute when Saiz eventually latched on to O’Kane’s through ball to Alioski to smash the ball home from close range.
City should have pulled a goal back in the next minute but Bobby Reid put a free header wide from Callum O’Dowda’s cross.
City gradually got going and the Robins were aggrieved when referee Oliver Langford failed to award them a 23rd-minute penalty when Gaetano Beradi caught and tripped Jonathan Leko.
But United continued to look menacing and Saiz wasted a potential opportunity to net a hat-trick when attempting a Hollywood finish from another counter attack which Fielding saved.
City then attempted to turn the screw but Leeds held firm to take a 2-0 lead into the break with Luke Ayling producing a fine block to keep out a strong drive from O’Dowda in the fourth minute of stoppage time.
Leeds were straight back on the attack after the break as Phillips headed over from an Alioski cross.
Reid then squandered another opportunity for City when losing his footing after breaking into the box before Jens Hegeler headed over from a free-kick.
But after Lasogga had seen a strong effort from 20 yards well saved, United squandered two fine chances shortly before the hour.
First Lasogga and Vieira worked the ball to Saiz whose sliding effort went wide with his hat-trick begging.
Then Aden Flint produced a crucial block to keep out a goalbound effort from Lasogga.
At the other end, Hegeler saw his free header lack direction allowing Lonergan to produce a good save.
But Leeds looked the more likely to score again and the Whites did exactly that in the 67th minute when Lasogga produced an excellent flicked header from a Phillips corner to beat Fielding all ends up.
Leeds were cruising to victory though the match ended in controversy after Berardi and Taylor were both shown straight-red cards after clashing in the 79th minute.
Leeds United: Lonergan, Ayling, Pennington, Cooper, Berardi, Phillips, O’Kane, Vieira, Alioski (Sacko 71), Saiz (Anita 83), Lasogga (Grot 89). Subs not used: Wiedwald, Jansson, Roofe, Hernandez.
Bristol City: Fielding, Wright, Flint, Hegeler (Magnusson 68), Bryan, Leko (Taylor 57), Brownhill, Pack, O’Dowda, Reid, Woodrow (Diedhiou 57). Subs not used: Steele, Vyner, Kelly, Eliasson.
Referee: Oliver Langford.
Attendance: 24,435 (3,711 away).