Former Spain boss Lopetegui was said to be on the brink of the sack after losing to Levante last week and he cut a disheartened figure on the Camp Nou touchline as Madrid was hammered in his 14th game in charge.
The irrepressible Suarez led Barca, which was without Lionel Messi, and capitalised on its rival's defensive failings to move back to the top of LaLiga.
Philippe Coutinho opened the scoring and Suarez doubled the advantage in a devastating first-half showing, but Marcelo's response shortly after the restart hinted Madrid might have some fight in it.
Luka Modric hit the post shortly after that, an inch more inside it probably would have levelled the scores, and Los Blancos started to gain a hold on the game.
But the introduction of Ousmane Dembele for Coutinho provided the pace Barca was lacking, and it soon paid dividends.
He was involved in the build to as Suarez executed a magnificent header, and the Uruguay star then pulled of a chipped finish to beat Thibaut Courtois before Arturo Vidal added the fifth from Dembele's cross to prompt a party in the stands as Lopetegui, whose side remains ninth, shuffled towards the exit door.
Barcelona swiftly identified the ball in behind as an effective route to goal and so it soon proved. Ivan Rakitic played in Jordi Alba, whose instant control gave him time and space to complete a 30-pass move and pick out Coutinho in the centre for a cool finish after 11 minutes.
Madrid's defensive frailties were again exposed on the half hour as Raphael Varane tripped Suarez in the area and, after a delay as the VAR (video assistant referee) instructed on-field official Jose Maria Sanchez Martinez to study the replay, the felled Barca forward tucked his spot-kick beyond Courtois.
The half-time introduction of Lucas Vazquez belatedly got Madrid further up the pitch and an improved spell was rewarded with Marcelo's goal, his third in as many games.
Modric's whipped effort against the foot of the post proved to be the end of Madrid's attacking threat as Barca turned on the style in a searing sprint to line.
Suarez turned home a Sergio Roberto cross with a stunning header and added a delicious chip over Courtois to complete his treble, before Vidal emerged from the bench to get in on the act with a late header at a raucous Camp Nou.
The numbers behind Lopetegui's tenure do not make for good reading. Starting with him leaving Spain in the lurch by agreeing to join Madrid on the eve of the FIFA World Cup, the 52-year-old coach seems to have made the wrong call at every turn. Madrid was torn apart at times in this game, and the same fate looks likely for his contract.
Nacho was seemingly selected as a defensive option on the right, able to keep Alba in check but offering little running the other way in the first half. But that decision merely invited Alba on and, far quicker than his opponent, he enjoyed a fine match.
From the white heat of the first Clasico of the season, the two clubs head to midweek Copa del Rey fixtures. Barcelona has a trip to Cultural Leonesa, while Madrid visits Melilla.