The Serbian seventh seed, 25, cruised through the first set without dropping a game in just 25 minutes, but was given a scare in the second set, with the 66th-ranked Russian's powerful forehand causing her plenty of problems.
Kudryavtseva led 5-3, but was unable to convert a set point on the Jankovic serve, and was subsequently unable to close out the set on her own serve.
Jankovic steamed to a 6-3 lead in the tiebreak, but a poorly judged forehand enabled Kudryavtseva to stave off the first match point, and she then saved a second.
However, a backhand error by Kudryavtseva handed Jankovic the match, 6-0, 7-6 (7/5), after 83 minutes and ended a six-match losing streak for the Serbian, who battled poor form in the latter part of 2010.
The 2008 Australian Open semi-finalist actually hit one less winner than the 23-year-old Russian, 18 to 19, but made eight fewer unforced errors (32 to 24) and fired down 10 aces.