Jeev is now nine-under through three rounds and seven adrift of the leader -- Lee Westwood (68) at 16-under India's other player in the field, Shiv Kapur missed the cut earlier after the second round.
Westwood retained his three-shot lead with a round to go, but his game was not as good as the second round when he shot 64. Yet Westwood was three clear of Ross Fisher who had a bogey-free 65 with seven birdies.
Jeev, who had his best result of 2011 at the same event, began with three birdies on the first three holes. He dropped a bogey on the fifth but quickly hit back with a birdie on the sixth. Thereafter, he had just one birdie on 13th and no more bogeys for a card of 68.
Swedish hopes of a third consecutive home win in this event after Richard S Johnson and Alex Noren's triumphs look best placed with Peter Hanson (69).
The Ryder Cup star had a difficult front-nine with a birdie and a double bogey. Then he holed a 15-footer at the 10th and had three more birdies for an inward 32 and an overall 69.
That left Hanson 12-under overall, four back with a round to play.
Noren continued his remarkable recovery from an opening 74 with a second successive 66 to move into fifth on 10-under, alongside compatriot Michael Jonzon.