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 were dismantled in his 14th game in charge.
The irrepressible Suarez led Barca, who were rampant even without Lionel Messi and capitalised on their rivals' failings to move back to the top of LaLiga.
Philippe Coutinho opened the scoring for the champions and Suarez doubled their advantage in a devastating first-half showing, with Marcelo's response shortly after the restart merely a small bump in the road.
Suarez executed a magnificent header and then a gorgeous dinked finish, before Arturo Vidal added the fifth to prompt a party in the stands as Lopetegui, whose side remain 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 instructed referee 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.
Luka Modric whipped an effort against the foot of the post, but that was the end of Madrid's attacking threat as Barca turned on the style.
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.
14 games, six wins, six defeats
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 World Cup, the 52-year-old coach seems to have made the wrong call at every turn. Madrid were torn apart at times on Sunday and the same fate will surely become of his contract.
