In the match played behind closed doors in the aftermath of the violence the city witnessed on Monday, the hosts ended with full points at the Sree Kanteerava Stadium.
Vineeth struck in the seventh minute for the home side when he tapped home from a close range.
Bengaluru had a slew of chances to extend their lead but, barring the lone strike, had little to show for their possession. The second leg will be played on September 21.
Tampines had their first corner of the match in the fifth minute, but could not work it to their advantage, spilling the opportunity after Nishu Kumar cleared Penna's kick.
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After a see-saw battle for the next seven minutes, Tampines had the second chance by earning a corner for the second time, but the kick was cleared by Juan Antanio.
Notwithstanding the loss of Mustafic, Tampines constantly attacked, but could not get the equaliser.
Jermaine Pennat spilled the third corner, which he took in the 24th minute, after his team-mate was found to be in an offisde position.
BFC had a wonderful opportunity to take a 2-0 lead in the 31st minute after Chhetri headed down Rino Anto's cross for Eugeneson Lyngdoh to attack, but the latter's shot was saved by Mohammad Izwan Bin Mahbud.
In the second half, Tampines had a very good chance of striking the equaliser but Billy Mehmet did not get enough power on his shot in the 50th minute.
Nishu, who put in a solid show down the left flank, was replaced by Lalchhuanmawia in the 53rd minute.
In the final lap of the match, BFC made some in roads. The first move was created by Vineet after he dribbled past a opposition player, but his shot was inches high of the woodwork.
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