The new Hearts boss saw Ryan Stevenson score twice in 12 first-half minutes before two goals after the break secured the Jambos' passage 5-2 on aggregate.
If Stevenson's first was fortunate, as Paks goalkeeper Norbert Csernyanszki fumbled his 34th-minute header, then the second was deserved as he netted from substitute John Sutton's pass in first-half stoppage time.
Andrew Driver marked his return from injury with a well-worked third after 50 minutes to finish the tie as a contest and Rudi Skacel celebrated his six-month contract extension with an easy fourth.
The rout rendered the advantage of Jamie Hamill's first-leg away goal in the 1-1 draw in Szekesfehervar irrelevant, even though Daniel Bode headed in an 89th-minute consolation.
A first victory of the season, ending an 11-match winless run, saw Hearts progress to Friday's 76-team play-off draw alongside Scottish Premier League champions Rangers and last season's runners-up Celtic.
It was the result and performance onlooking owner Vladimir Romanov, who has tasked Hearts with winning the SPL this season, would have wanted - but they will not be playing Paks every week.
Two days following his appointment in place of Jim Jefferies, ex-Sporting Lisbon boss Sergio named a team showing two changes from the side which began last Sunday's 1-0 loss to Dundee United, with Driver back from injury and Stevenson called up to start.
Continual fouling by Paks allowed Danny Grainger to find his range at set-pieces, where Hearts were always dangerous.
After Stevenson's near miss from an Ian Black free-kick and the departure of his strike partner Stephen Elliott after 13 minutes, Hearts went in front from Grainger's delivery.
Former Ayr utility man Stevenson rose highest to meet Grainger's inswinging corner six yards out, but Csernyanszki should have done better than the weak left hand which allowed the ball to bounce into the net.
Grainger fired narrowly wide from a free-kick which went under the Paks wall soon after before Istvan Sipeki dragged an effort wide after 45 minutes for the visitors.
The counter-attack saw the hosts go two up as Hearts goalkeeper Marian Kello pumped the ball down the other end. John Sutton, on for Elliott, shaped to shoot before playing a neat pass for Stevenson to net from a narrow angle on the left of the area.
Hamill came on at half-time in place of captain Marius Zaliukas, who apparently was unable to recover from a knock sustained at the end of the first half.
Hearts' reshuffle, with Adrian Mrowiec moving into defence and Hamill lining up in midfield, was almost immediately exposed as Bode dribbled through and was allowed to shoot just wide.
But five minutes into the second half Hearts netted a third goal to all but finish the tie.
Stevenson played the ball into Sutton with his back to goal and the striker laid it off for Driver to net right footed in off the bar.
It was the first time the winger, who has been plagued by injury in recent seasons, has scored since March 2010.
Csernyanszki saved with his legs from David Templeton soon after, before Black headed over from a long Grainger throw-in.
Driver made way for Skacel, who earlier in the day agreed a new deal until the end of January, after 63 minutes before Black curled the ball over from 25 yards as Hearts maintained the pressure.
Laszlo Bartha's snap-shot was then blocked by Grainger as the hosts had to defend and Kello was called into action after Bode raced through, the Slovakian goalkeeper spreading himself to deny the Paks striker.
It was an example Csernyanszki failed to follow moments later, when he palmed Templeton's stinging drive into the centre of the penalty area for Skacel to prod in.
Ryan McGowan tumbled over as he surged into the box, with appeals for a foul waved away, before a Stevenson shot ricocheted off the Hearts full-back and just wide.
Kello was beaten when Bode headed in Tibor Heffler's free-kick, but it was little consolation for Paks as Hearts progressed.