Paulo Sousa's side entered the match bottom of the group, but level on three points from three games with Liverpool and Ludogorets.
The Bulgarian side beat 10-man Basel with a stoppage-time winner on matchday three, but there was to be no repeat as the hosts cruised to a vital three points.
Breel Embolo and Derlis Gonzalez punished poor defending with first-half goals before Shkelzen Gashi and Marek Suchy underlined the gulf between the two sides after half-time.
With Real Madrid out in front in Group B, the battle for second place is now finely poised ahead of Basel's trip to Anfield on matchday five.
Both sides started without their respective match-winners from earlier in the campaign with Yordan Minev suspended for Ludogorets and Marcos Streller still out injured for Basel.
And the visitors' intentions were clear from the first whistle as they packed men behind the ball, delaying and disrupting the game whenever possible.
That approach succeeded in stifling the Swiss champions and Ludogorets even created an unlikely chance of their own with a 26th-minute counter attack, only for Dani Abalo to panic and slice over the bar at the back post.
But the resistance was broken in the 33rd minute and for all their committed defending, Ludogorets were undone by a single throughball from Fabian Frei.
Embolo's run through the middle was not tracked and the 17-year-old took full advantage, shrugging off the challenge Georgi Terziev when it eventually arrived, before beating Vladislav Stoyanov with from 12 yards.
Ludogorets did not learn from their mistake and Gonzalez profitted from similarly slack defending in the 41st minute.
Frei was once again afforded time and space to pick out the Paraguayan, who advanced on goal before applying a cool finish past a static Stoyanov.
Mihail Alexandrov was thwarted in his attempts to reduce the deficit in first-half stoppage time, with Tomas Vaclik, in stark contrast to his opposite number, racing off his line to clear the danger.
The confidence was coursing through the Basel team as Embolo and Gashi tried their luck from distance early in the second half.
The pair then combined to put the result beyond doubt after 59 minutes as Embolo delievered a wonderful cross from the right that Gashi merely had to tap in from three yards.
Gashi looked like helping himself to another in the 65th minute, after more fine build-up play from the energetic Embolo, until a fantastic block by Brayan Angulo.
But Angulo's team-mates failed to match that level of defending from the resulting corner and Suchy was allowed to convert a loose ball from close range.
Embolo could have added a fifth in the closing stages, but was denied by Stoyanov and instead the loudest cheer of the night was reserved for home favourite Marcelo Diaz.
The Chilean came on to replace Frei and make his first European appearance since receiving a three-match in last season's Europa League quarter-final against Valencia, rounding off a fine night for the home side.
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