A day after boss Nuno Espirito Santo signed a new three-year contract, there was more reason to cheer as the Blades were blown away by a brilliant first-half display.
Jimenez got the ball rolling with a superb strike in the third minute before Saiss, in the sixth, made it two with an unstoppable header.
More of a cagey second half but, ultimately, a promising start as Nuno looks to take Wolves to the next level.
Considering Wolves only had a few full days of training together ahead of this season-opener at Bramall Lane, the way they came flying out of the traps was remarkable.
Having struggled so often to score in first halves last season - only coming up with 14 first-half goals in 2019/20 and leading just seven times at the break - it was so refreshing to see them give themselves an early cushion.
Chris Wilder's outfit were taken aback by the speed of Wolves' play in those opening minutes. They blinked and Nuno's lot were 2-0 up.
Jimenez - again, truly one of the best strikers around - displayed his prowess with a finish of the highest order. Then Saiss, who was magnificent from the first whistle to the last, headed home.
Aside from the goals, Wolves played some very slick football, with Daniel Podence and Pedro Neto also impressive in attack.
Fernando Marcal slotted in seamlessly at left-wing-back, too. There were so many positives for Nuno to take from it.
Yes, it was not plain sailing in the second half and the right-wing-back position needs addressing before the end of the window, but Wolves have time.
For now, it has been an excellent couple of days for Nuno and everyone else connected with Wolves. A very encouraging performance.
Nuno gave new signing Marcal the nod at left-wing-back, while Podence and Neto were supporting Jimenez in an attack-minded XI.
Interestingly, Joao Moutinho and Leander Dendoncker were the two in central midfield while Adama Traore - following Matt Doherty's sale to Tottenham - was on the right flank.
That meant Ruben Neves was on a strong-looking bench along with club-record £35million signing Fabio Silva and the other fresh face, Vitinha.
United, meanwhile, sprung a slight surprise as Jack O'Connell started in defence despite missing all of their pre-season friendlies.
Wolves, of course, did not have any warm-up games at all given the quick turnaround from the Europa League quarter-final loss to Sevilla, so there was potential for a slow start to the clash.
But there were no signs of rustiness as they stunned the Blades and raced into a two-goal lead.
A rip-roaring counter-attack saw Jimenez sublimely open the scoring, just three minutes in.
Moutinho cleverly picked out the run of Podence, who left John Lundstram for dead before finding Jimenez, who blasted the ball beyond the helpless Aaron Ramsdale on his 100th game in gold and black. Such an emphatic finish. New season, same superb Jimenez.
And come the sixth minute, Wolves had doubled their advantage.
Neto's inviting delivery from a corner was powered home by Saiss, rising highest and coming up with a bullet header. Again, there was nothing Ramsdale could do.
So often so cagey in first halves last season, especially post-lockdown, Nuno's charges bucked that trend.
They were showing signs of evolution in their play, too. It was all very fluid.
At times, it looked like a 3-4-3 formation, and then others a 3-5-2. Podence and Neto, in particular, were popping up in different positions and constantly asking questions.
Dendoncker was keen to burst forward, as was Traore down the right, while Marcal - in a similar vein to the injured Jonny Castro Otto - looked solid on the left.
The hosts, frankly, did not know what had hit them.
Jimenez sparked fear into United every time he had the ball. He forced Ramsdale into a smart save after scoring and saw a left-footed curler sail narrowly off target.
A delightful turn past John Egan displayed the level of confidence he was playing at, too.
Wolves were not majorly troubled by the Blades before the break and, in truth, it was pretty much a perfect half for Nuno's pack.
The second period started with the home side piling on the pressure, though. Enda Stevens managed to get in behind Traore but lashed his effort from an acute angle over.
That was a bit of a let-off, and something Nuno was not pleased with. John Fleck soon smacked the woodwork, and then George Baldock's attempt sailed just past the far post.
Things were becoming a bit more tricky for Wolves, but they were doing enough with Conor Coady - boosted by his recent England debut - organising the backline and making timely blocks.
Saiss almost managed his second of the evening as he met Moutinho's corner with another towering header - this one bouncing a couple of yards wide.
Back at the other end, Lundstram could not get on the end of Billy Sharp's driven ball across goal.
It was all a bit frantic, with United finding joy down the left - so much so that Nuno made a change.
Oskar Buur came on for his top-flight bow, for Neto, which allowed Traore to move further forward. Wolves almost made it three straight afterwards as well.
Saiss, for a few seconds, transformed from a centre-half into a centre forward as he latched onto Jimenez's pass and saw his shot tipped onto the post by Ramsdale. Jimenez, having to react quickly, put it wide on the rebound.
The Blades, though, were still knocking on the door, and Neves was introduced to try to tighten up the midfield for the last quarter of an hour.
Vitinha was also introduced, for Moutinho, as Wolves calmed down and saw out the win. Traore and Jimenez went close late on while Saiss cleared an Oli McBurnie header off the line. In the end for Wolves, two goals and three welcome points to kick off 2020/21.
Sheff U (3-5-2): Ramsdale; Basham (McGoldrick, 71), Egan, O'Connell; Baldock, Norwood (Berge, 76), Fleck, Lundstram, Stevens; McBurnie, Sharp (c)
Subs: Foderingham (gk), Ampadu, Jagielka, Osborn, Burke
Wolves (3-4-3): Patricio; Boly, Coady (c), Saiss; Traore, Moutinho (Vitinha, 86), Dendoncker, Marcal; Neto (Buur, 69), Jimenez, Podence (Neves, 76)
Subs: Ruddy (gk), Vinagre, Jota, Silva
Goals: Jimenez (3), Saiss (6)
Referee: Mike Dean (Wirral)