The left back popped up with a late equalizer to rescue a draw after a below-par performance from Claudio Ranieri's side.
Vincent Aboubakar had given the home side an early lead before Valere Germain equalized for the principality club on the stroke of halftime.
Jeremie Aliadiere thought he’d won it for the home team with 15 minutes remaining, but Kurzawa’s late strike ensured the club earned a share of the spoils.
Lorient started much the better side and was almost ahead inside four minutes when Aboubaker beat Moutinho in the area before lifting the ball over Monaco keeper Danijel Subasic and onto the top of the crossbar.
Aboubaker quickly went one better, however, giving the home side a deserved lead with a header from Kevin Monnet-Paquet's cross to score his 12th of the season on 14 minutes.
Monaco was completely outplayed in the opening stages and Alain Traore and Aboubaker wasted further chances to double the home side’s lead.
The visiting team's diamond formation had earned it a 2-0 victory over Marseille last week but Claudio Ranieri was forced to make a tactical substitution after just 26 minutes, withdrawing Mounir Obbadi for Lucas Ocampos and switching to a flat 4-4-2.
New signing Dimitar Berbatov isn't yet available for Monaco and the Bulgarian will do well to oust Germain, who eventually equalized for the away side with a right-footed finish from just outside the box.
James should then have put Monaco ahead right on the stroke of halftime, but fired well over the bar from six yards after doing well to cut inside the defender onto his left foot.
Lorient started the second half well and nearly restored the lead on 50 minutes when Subasic had to rush from his line to block Aliadiere’s effort, before Bruno Ecuele Manga saw his follow-up stopped on the line by Andrea Raggi.
Alliadiere put Lorient back into the lead 15 minutes later. Boueyan’s shot was saved by Subasic but his parry found Aoubcar, who squared for Alliadiere to fire the ball into an empty net.
Kurzawa equalized with just four minutes left when he controlled Ocampos’ low cross inside the box before placing his left-footed shot wide of Lorient keeper Florent Chaigneau and into the far corner.
Germain almost pinched all three points right at the end but saw his shot flash just wide.
The draw means third-placed Lorient remains eight points behind Monaco in second.