The Toffees had trailed 1-0 at half-time in horrendous conditions in Minsk, but produced a spirited second-half display to further strengthen their position at the top of Group I.
David Moyes was without nine first-team regulars for the trip to Belarus and named a makeshift side with Tony Hibbert forced into an unaccustomed central defensive position and several youngsters populating the bench.
He made five changes to the side that defeated Portsmouth in the Premier League on Saturday but they still threatened first when Diniyar Bilyaletdinov rattled the post with a 20-yard shot after good set-up play from Ayegbeni Yakubu.
But BATE slowly found their feet and took the lead in the 16th minute through Dmitry Likhtarovich.
The home skipper latched on to a loose ball 35 yards from the Everton goal and let fly with a shot which left goalkeeper Tim Howard with no chance.
Everton should have been on level terms just before the half-hour mark when Yakubu found Cahill in the box but the Australia midfielder was unable to get a clean shot away and it was blocked out for a corner.
Leon Osman and Jo both wasted half-chances in the early stages of the second half in what was now driving rain.
But the Toffees didn't have to wait too much longer to restore parity as the unmarked Fellaini was allowed to meet Leighton Baines' deep free-kick and find the opposite corner of the net with a simple 68th minute header.
It was 2-1 nine minutes later when Cahill came up with a moment of real quality.
Baines' pass was flicked on by Bilyaletdinov for Cahill to race on to. The midfielder then clipped the ball over the head of the advancing keeper to put his side into a lead they would not relinquish.
BATE pushed forward in the closing minutes and threatened on several occasions after refusing to throw in the towel.
However, some resolute defending ensured Howard's goal was not breached again and left the Merseysiders within touching distance of the knockout stages of the competition.