The towering Argentine, who had not played on clay since last year's French Open, struggled to find his feet in the opening set as Dolgopolov dictated play and produced some fine drop-shots.
As the sun set over the Mediterranean, Del Potro started to play deeper and took advantage the Ukrainian's wayward shots to wrap up the next two sets for victory.
"It was really tough to start playing on clay against him. You always try to play long rallies to feel the clay, to settle your play," fifth seed Del Potro said.
"He plays fast, he comes to the net often ... After one hour and a half, I felt better," added the world number six, who had not planned to come to Monte Carlo but was given a wildcard.
French number one Jo-Wilfried Tsonga also made it to the last 16 thanks to an authoritative 7-6 6-2 victory over Nikolay Davydenko.
Sixth-seed Tsonga trailed 0-2 in the first set but quickly regrouped and whipped up the roaring crowd by charging to victory.
World number nine Richard Gasquet completed a successful day for the French favourites as he showed no ill effects of the ankle injury that kept him out of the Davis Cup quarter-finals and completed a 6-1 3-6 6-1 win over countryman Benoit Paire.
Serbia's Janko Tipsarevic was the only top-10 player to suffer an early exit with a 7-6 6-1 defeat by Bulgarian Grigor Dimitrov.
World No.10 Tipsarevic, who has never made it past the third round at the claycourt Masters series event, extended his poor win-loss record to 2-6 since he reached the Australian Open last 16 in January.
The 21-year-old Dimitrov, regarded as one of the most gifted prospects on the tour, recovered from a break down in the first set then easily prevailed in the tie-break, wrapping it up with a subtle backhand pass.
The 34th-ranked Bulgarian then cruised to his first win of the season over a top-10 player to reach the last 16.
Results from the Monte Carlo Masters
5-Juan Martin Del Potro (Argentina) beat Alexandr Dolgopolov (Ukraine) 1-6 6-4 6-3
6-Jo-Wilfried Tsonga (France) beat Nikolay Davydenko (Russia) 7-6(3) 6-2
16-Philipp Kohlschreiber (Germany) beat Pablo Andujar (Spain) 6-2 2-6 7-6(3)
7-Richard Gasquet (France) beat Benoit Paire (France) 6-1 3-6 6-1
Grigor Dimitrov (Bulgaria) beat 8-Janko Tipsarevic (Serbia) 7-6(3) 6-1
Ernests Gulbis (Latvia) beat John Isner (U.S.) 4-6 6-2 6-3
10-Nicolas Almagro (Spain) beat David Goffin (Belgium) 2-6 6-1 6-3
14-Juan Monaco (Argentina) beat Martin Klizan (Slovakia) 6-4 6-0
Albert Montanes (Spain) beat Gael Monfils (France) 6-2 2-6 6-4
Kevin Anderson (South Africa) beat Jerzy Janowicz (Poland) 7-5 7-6(9)
Jurgen Melzer (Austria) beat Jesse Huta Galung (Netherlands) 1-6 6-3 6-2
Marcel Granollers (Spain) beat Victor Hanescu (Romania) 6-7(8) 6-3 6-1
Florian Mayer (Germany) beat Benjamin Balleret (Monaco) 5-7 6-1 6-2
Marinko Matosevic (Australia) beat Fernando Verdasco (Spain) 7-5 6-3
Albert Ramos (Spain) beat Radek Stepanek (Czech Republic) 6-3 6-3
13-Stanislas Wawrinka (Switzerland) beat Denis Istomin (Uzbekistan) 6-3 7-5