The Saints striker set up debutant Steven Robb for the home side's 18th-minute goal which opened the proverbial floodgates.
Mehmet and Andy Dorman scored before the break as the Paisley side ensured they would not suffer similar embarrassment to last season when they lost in the same competition to Third Division East Fife.
Two goals in two minutes soon after the interval earned Mehmet the match ball before he went off to a standing ovation from most of the 1,747 supporters inside Love Street, replaced by Dennis Wyness.
Substitute Craig Dargo, on at the same time for Jim Hamilton, made it six and Gary Mason added the seventh against the sorry-looking Sons.
Surprisingly, given the obvious gulf between the two sides, it took a while for the Scottish Premier League outfit to get in to their stride.
Mehmet had a couple of efforts, one high and wide from distance, the other trundling in the arms of David McEwan from 16 yards out.
However, following the first of several impressive first-half passages of play, Mehmet split the Dumbarton defence allowing Robb to speed through the centre and dispatch his shot past McEwan from the edge of the box.
In the 22nd minute, Mehmet should have got on the scoresheet himself when he took a Franco Miranda pass in his stride 16 yards from goal but fired his angled shot wide of the far post.
In the 33rd minute, following a Dorman corner which made its way to the back post, Hamilton drove just over the bar.
But two minutes later Mehmet grabbed the home side's second with a close-range drive after Dumbarton defender Michael O'Byrne made a mess of clearing a Miranda cross from the left.
Dumbarton were all over the place at the back and six minutes from the interval Dorman worked a clever one-two with Hamilton before dinking over McEwan from 12 yards.
The second half offered little respite for the bedraggled visitors.
Five minutes after the break, Mehmet grabbed his second when he turned O'Byrne just outside the six-yard box and slotted Hamilton's pass home.
Two minutes later, he completed his hat-trick when he headed Garry Brady's cross past the helpless McEwen.
Dargo replaced Hamilton, with Wyness on for Mehmet, and the traffic continued towards the Dumbarton goal.
In the 70th minute, in the face of constant pressure, McEwen pulled off a fine save down at his left-hand side from Wyness' low drive.
But Saints were not to be denied, with Dargo and Mason stretching their lead further within a two-minute spell as Dumbarton imploded.