Bilate struck in the 82th minute, meaning Jackie McNamara's side earned a deserved 1-0 win after they had done most of the pressing in the second half.
The summer signing hit a powerful drive from 25 yards which left Well keeper Dan Twardzik spinning as it found the top corner to help the Taysiders build on their opening-day win over Aberdeen.
Both teams had won on the first day so went into this match hoping to prove their form was true. It was Motherwell who had the first chance as Josh Law tried his luck from distance only to see his effort come back off the crossbar with United keeper Radoslaw Cierzniak beaten.
United attempted to hit back in the 12th minute when Ryan Dow's pass found Gary Mackay-Steven, but he shot wide of the target.
At the other end, Jack Leitch's in-swinging corner picked out Fraser Kerr, only for his header to glance a couple of yards over the bar.
Motherwell began the second half with David Ferguson replacing Zaine Francis-Angol in the hope they could step up a gear.
But within minutes of the restart Ferguson found himself being booked by referee Kevin Clancy for hauling down Mackay-Steven.
United threatened on the hour mark with Stuart Armstrong threading a pass through for Dow, but Ciftci could not connect with his header.
Ciftci was involved again in 65 minutes as he turned swiftly inside the box to unleash a shot which was blocked by Well defender Stephen McManus.
There were appeals for a penalty from the hosts but Clancy waved play on amid suggestions it had hit McManus' hand.
Soon after, Bilate made his first home appearance for United by coming on as a substitute for Watson.
In the 67th minute, though, Motherwell had the ball in the net after Lee Erwin was put clean through on goal to slot past Cierzniak, yet he was rightly flagged for being offside.
Paton should have put the hosts ahead in the 80th minute but fired over the bar with the goal at his mercy.
However, just as the game looked to be heading for a stalemate, Bilate popped up to make himself the hero for the hosts, who maintained their impressive start to the new campaign.