Antonio Conte's men took a two-goal lead into the second leg, which was played behind closed doors due to concerns over the coronavirus outbreak in Italy.
Cauly profited from some sloppy defending to halve the deficit with 26 minutes played, but Cristiano Biraghi squeezed in a deflected effort from an angle after being played in by Eriksen to level things up on the night.
Romelu Lukaku edged the Italian club ahead with a strange goal just before half-time and competition favourites Inter saw things through to book a place in Friday's last-16 draw.
Lukaku, who had also scored a late penalty in last week's first leg after Christian Eriksen's opener, failed to beat Plamen Iliev from a one-on-one five minutes into this return match.
Bulgarian champions Ludogorets gave themselves some hope when Cauly held off Diego Godin and fired past Daniele Padelli at his neat post via the underside of the crossbar.
But Inter restored its two-goal aggregate margin 188 seconds later as Biraghi's drilled effort from the left went in via a deflection off visiting defender Georgi Terziev.
Lukaku then atoned for his earlier miss by scoring a bizarre fourth goal of the tie for Inter, with the ball ricocheting off his body while on the floor after his initial header was saved.
Alexis Sanchez played a big part in that goal and struck the frame of the goal with a 25-yard drive early in the second half.
Lukaku was also denied by a mixture of Iliev and the post five minutes later but, a near-miss from substitute Mavis Tchibota aside, Inter had little trouble in seeing out a second victory on the bounce ahead of Sunday's trip to Juventus.