Leonardo Jardim's side lost 2-0 at home to Arsenal on Tuesday, but their 3-1 victory in the first leg proved enough to see them through on away goals.
It is the first time Monaco have reached the last eight since 2004, when they went on to make the final, but domestic matters come to the fore now, with a trip to Reims on Sunday.
"Now we must turn the page and focus on the next game in the championship, and aim to be playing back in the Champions League next year," defender Aymen Abdennour said after the Arsenal game.
Monaco sit fourth in the table, one place and four points adrift of the final qualifying berth for European football's premier club competition, although they do hold a game in hand over the leading trio.
Leaders Lyon, Paris Saint-Germain and Marseille all dropped points last time out, with the champions' 3-2 defeat at the hands of Bordeaux meaning they were unable to gain ground on their two title rivals, who played out a goalless draw.
Zlatan Ibrahimovic's foul-mouthed tirade stole the headlines after PSG's defeat, and the Sweden international could face sanctions when the Ligue de Football Professionnel Disciplinary Commission investigates his comments during its session on Thursday.
Marseille's Dimitri Payet may also find himself in hot water after appearing to rant in the direction of the officials' dressing room following Sunday's stalemate at home to Lyon.
Second-placed PSG, two points behind Lyon, could temporarily go top of the table when they host Lorient on Friday.
Victory would see Laurent Blanc's men climb to the summit, but Lyon will have the chance to reclaim their spot when they welcome Nice 24 hours later, while Marseille - two points further back - travel to struggling Lens on Sunday.
Following crowd trouble at Bastia's 2-1 home victory over Nice earlier this month, the Corsican club have been forced to play their home matches on neutral grounds on the mainland, and behind closed doors.
Saturday's meeting with Guingamp will therefore be held at Istres' Stade Parsemain.
Struggling Toulouse host Bordeaux in the Derby de la Garonne, while bottom club Metz are hoping to finally end a 17-match winless run when they travel to Caen.
Elsewhere, Saint-Etienne will seek to keep their Champions League qualification hopes alive when they play host to Lille, while Evian welcome Montpellier and Nantes visit Rennes.
