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The fourth-seeded Serbian will play 150th-ranked Croatian qualifier Mirjana Lucic in the second round, but the 2008 US Open runner-up nearly suffered her first opening-match exit at a Grand Slam event since the 2005 French Open.
Jankovic, 25, received a spirited fight from an 18-year-old Romanian whose only prior Grand Slam match was a first round loss at this year's French Open.
Halep and Jankovic exchanged breaks to open the third set, then traded holds until the ninth game, when Jankovic sent a forehand beyond the baseline to surrender a break, leaving Halep ahead 5-4 and serving for the match.
Jankovic fired a forehand wide on a break point and began talking to herself in frustration, then hit a forehand winner on a second break chance to equalize at 5-5, leaving Halep tapping her racket and slapping her leg in frustration.
Halep surrendered the last game at love and Jankovic escaped her worst US Open showing just a year after a second-round ouster despite 48 unforced errors to 36 for Halep. Each fired 22 winners.
Jankovic, whose only Grand Slam final was her 2008 US Open loss to Serena Williams, and Danish top seed Caroline Wozniacki can move past injured Williams to claim the world number one spot by capturing the US Open title.
Wozniacki, who lost to Kim Clijsters in last year's US Open final, begins her quest for the crown late Tuesday against US wild card Chelsey Gullickson.
Chinese eighth seed Li Na was set to play Kateryna Bondarenko and Russian 14th seed Maria Sharapova was to meet Australian Jarmila Groth in other later first-round matches.
Jankovic went 1-3 in US Open tuneup events. Her only title of the year came in March on the Indian Wells hardcourts, where she beat Wozniacki in the final.
Jankovic broke Halep on her way to a 3-0 lead in set one. The Romanian teen broke back in the fifth game and the two exchanged breaks until Jankovic took the first set after 42 minutes when Halep netted a forehand.
But Jankovic, who had 15 unforced errors to seven by Halep in the set, was having trouble and about to face worse.
Halep began the second set by breaking Jankovic, then held to end the run of service breaks at seven in a row. Halep fired a backhand winner to the back corner on her next break chance and took a 3-0 edge.
Halep fought through six deuces before holding again for a 4-0 lead and stretched the lead to 5-1 before Jankovic rallied, holding twice around a break at love before Halep held again to force a third set.
