In her first professional match in almost half a year, world number two Halep, who skipped last week's Palermo Open and will not make a decision on whether to play the US Open or not until after the tournament in the Czech Republic, was made to work hard for victory by the unseeded Hercog.
The Romanian picked up an early break and responded to dropping her serve in game three of the first by reeling off the next four in succession to take the set.
When Halep started the second set by breaking Hercog the writing appeared to be on the wall, but the Slovenian world number 46 fought back valiantly to force a decider in emphatic fashion.
Halep was unable to convert any of the six match points that came her way after breaking for a 5-3 lead, with Hercog clinging on to take the match to a third-set tie-break.
After failing to consolidate an opening mini-break, Hercog's resistance was finally broken after two hours and five minutes when a forehand into the net sent top seed Halep into the second round.
Wildcard Barbora Krejcikova ensured Halep's next match would not be an all-Romanian affair by beating Patricia Maria Tig 6-4 6-3.
Second seed Petra Martic defeated Varvara Gracheva 7-6 (7-2) 6-3 and third seed Elise Mertens needed three sets to overcome Jasmine Paolini 7-5 4-6 6-3.
A number of other seeded players were not as successful. Ekaterina Alexandrova (5) fell to qualifier Lesia Tsurenko, while Anastasia Pavlyuchenkova (6) went down 7-5 6-2 to Arantxa Rus, and Anastasija Sevastova (9) lost 6-2 6-2 to Irina-Camelia Begu.
Linette took just 31 minutes to win the first set in Rome, but world number 10 Kvitova responded superbly to win 1-6 6-0 6-2.
Two-time grand slam champion Kvitova racked up 16 unforced errors in the first set, yet rallied like the great competitor she has proved to be for so long to ensure she will face Vera Zvonareva - conqueror of Christina McHale - in the second round.
The 11th seed from the Czech Republic said: "The first set I started very badly. I was missing almost everything. On the court... [the ball] was bouncing everywhere as well. A little bit windy. I just had to get used to probably everything.
"The beginning of the second set when finally I held my serve and broke her serve, I think since then it was much, much better."
French Open champion Iga Swiatek advanced when Alison Riske retired at 5-4 down in the first set due to a recurrence of a left foot injury.
Swiatek, who will take on Sloane Stephens or Madison Keys in the second round, made 18 unforced errors but had won four games in a row from 4-1 down when the American was unable to continue.
Coco Gauff battled past Yulia Putintseva 7-5 4-6 6-4 to set up a meeting with 17th seed Maria Sakkari - who fended off qualifier Polona Hercog 7-6 (8-6) 3-6 6-2.
Amanda Anisimova will do battle with Elina Svitolina, a winner of this tournament twice, after defeating Wang Qiang 7-5 6-3. Jessie Pegula, Jennifer Brady and Petra Martic were among the other winners on day one.
Raducanu, who was the surprise US Open champion in September after not dropping a set throughout the tournament, appeared at just her fourth tour-level tournament and came to Cluj as third favourite.
The 18-year-old had lost opening-round matches at previous events in Nottingham, San Diego and Indian Wells.
However, world number 23 Raducanu finally recorded her first victory as she overcame a first-set scare to triumph over Hercog 4-6 7-5 6-1.
In the opening match of Tuesday's fixtures, sixth seed Marta Kostyuk cruised past Bernard Pera 6-3 6-4 to secure her last-16 berth.
Irina-Camelia Begu, ranked one behind Kostyuk for the tournament, did not enjoy similar fortunes as Romanian compatriot Irina Bara triumphed 7-6 (7-2) 0-6 6-4.
Jaqueline Cristian survived a marathon slog against Kaja Juvan as she came from a set down to win 3-6 7-6 (10-8) 7-6 (7-3) and book a second-round meeting with Ajla Tomljanovic.
Number one seed Simona Halep – who departed at the quarter-final stage of the Kremlin Cup last week – plays her first game on home turf on Wednesday against Elena-Gabriele Rusa and could meet Raducanu in the semi-final stages.
Meanwhile, Liudmila Samsonova, the third favourite at the Courmayeur Open, cruised past Stephanie Wagner 6-3 6-0 while seventh seed Jasmine Paolini comfortably dispatched Martina Di Giuseppe 6-4 6-1.
Shuai Zhang, who is the third seed in Italy, also enjoyed a routine victory as she eased past Jessica Pieri 6-1 6-3 after fourth favourite Petra Martic crashed out 3-6 4-6 to Saisai Zheng.
