Two second-half goals from Marouane Fellaini and Tim Cahill saw a depleted Everton side combat the weather and win at BATE Borisov.
The Toffees were without nine first-teamers for the Europa League clash including Joseph Yobo who pulled out late on with a thigh injury.
The Nigerian's absence meant Tony Hibbert had to move to centre-half with midfielder Dan Gosling filling in at right-back to make up a patched up backline.
In driving rain, which worsened throughout the match, the visitors began the brighter and could have taken the lead when Russian winger Diniyar Bilyaletdinov saw his thunderous left footed strike come back off the far right-hand post.
However, David Moyes' men found themselves behind when Dmitry Likhtarovich thundered in an unstoppable effort from all of 35 yards out on 16 minutes.
Everton began the second period much brighter and were back on level terms when Fellaini headed in a free-kick when left completely unmarked on 68 minutes.
The Toffees grabbed the winner nine minutes later when Cahill cleverly finished over the onrushing Sergey Veremko in the BATE goal.
Everton almost took but Bilyaletdinov was left cursing his luck. The Russian winger saw his powerful 20-yard shot come back off the post after Yakubu Aiyegbeni had carved out the opening.
BATE responded and made the breakthrough with Likhtarovich scoring a goal out of the blue, latching onto a loose ball 35 yards out and thumping a shot high over goalkeeper Tim Howard.
That got the home crowd going and gave BATE an early boost having lost their opening game to Benfica.
Everton almost drew level in the 26th minute when Yakubu released Cahill inside the six-yard box - but the Australian midfielder failed to get a decent connection and his effort was deflected wide.
Everton again pushed forward in the 37th minute and Fellaini laid the ball back to Leon Osman, skipper on the night, but his effort was blocked.
BATE, who drew twice with Juventus in the Champions League last season, then found themselves on the back foot though three minutes later.
It needed a timely clearance from Igor Shitov to prevent Yakubu from racing clear on goal, then Osman got clear inside the area after 54 minutes but his shot was scrambled to safety.
Three minutes later Jo peeled away from his marker and got on the end of a cross from Yakubu. However his soft header was easily taken by the goalkeeper.
BATE were not content to sit on their lead, however, and kept pushing forward at every opportunity.
Aleksandr Pavlov steadied himself outside the area in the 61st minute but sent his shot wide of the target, before Yakubu was hauled back for moving into an offside position three minutes later following a pass from Gosling.
The Nigerian questioned the decision and it looked as though he had a point.
Everton drew level in the 68th minute when BATE failed to deal with a Leighton Baines free-kick. Fellaini got on the end of it at the far post and found the corner of the net.
Everton went ahead in the 77th minute after a swift counter-attack that stunned the hosts. Bilyaletdinov headed the ball on and Cahill sprung the offside trap before hoisting a shot over the goalkeeper.
It was a remarkable turnaround by Everton and testimony to their fighting spirit after such an indifferent first half.