Rublev spurned a chance to seal the match when he fired a forehand narrowly wide on his own serve late in the deciding set.
Nadal had a much easier match but also struggled to put the finishing touches on his win.
Djokovic hailed Thiem as one of the best three players on clay in the world ahead of their meeting and the 24-year-old will now likely have the chance to contend with the undisputed king of the dirt, should he beat Khachanov later on Thursday.
"I was 10 centimetres from being out of the tournament".
"It was a battle until the last point, a good test", said Djokovic, who will next face fifth-seeded Dominic Thiem of Austria.
"It's a disgusting draw for sure, from the first round on", conceded Thiem.
World number seven Thiem laid down a marker in the opening stages, holding to love before breaking the Djokovic serve with a clean forehand victor.
Earlier on Tuesday night games Alexander Zverev, Grigor Dimitrov and Dominic Thiem all came from a set down to win their second-round matches at the Monte Carlo Masters.
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Austrian Thiem is one of the best claycourt players on tour, having reached the French Open semi-finals in each of the last two seasons, and he was impressed by his vanquished opponent's performances in Monaco.
The Croatian saved two match points while trailing 5-3 in the second set and recovered from 40-0 down on Djokovic's serve before saving four match points in an epic 10th game.
The first game set the tone for the rest of the match, with Djokovic eventually holding after nine minutes of bruising baseline rallies that included a missed break point for Coric.
"Under the circumstances and considering I haven't played too many official matches, I thought I played well". "I'm really happy with the way things are working out right now with the elbow".
The 31-year-old Spaniard wrapped up a comfortable victory in the Monaco sunshine with another break as Bedene hit a forehand into the net. "In a match of power tennis that featured 15 service breaks, 6'6" Zverev overcame his 6'5" compatriot, Jan-Lennard Struff, in a 6-4, 4-6, 6-4 win over two hours and three minutes.
"I made some small minor changes that actually in our world are quite significant", he said.
Earlier, French number one Lucas Pouille blew a one-set lead to be dumped out by Zverev's older brother Mischa in a third-set tie-break.
The former world number four, who has struggled over the previous year with a right wrist injury, followed up his first-round victory against Tomas Berdych with a solid display on centre court. After dropping his serve to trail 4-1, Bautista Agut cleared a ball from the back of the court just as a ball boy was sprinting across.
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