There was little sign of the rout which was to come in the final 25 minutes as the Ports led the Mallards 3-0 at half time through goals from Twigg, Braniff and Peter McMahon, who opened the scoring with a simple tap-in after Darren Murray's effort had hit the bar.
Twigg followed that strike with a lobbed effort in the 24th minute, before Braniff's header gave the hosts a three goal cushion at half-time.
Murray got himself on the scoresheet in the 65th minute before Dwayne McManus was sent off for the visitors for a second yellow card.
Braniff fired in a superb long-range effort soon after, before McMahon and Twigg added to their respective tallies.
It got worse for the Co Fermanagh side in the 81st minute as they were reduced to nine men when Steve Feeney was sent off for a cynical hand-ball near the goal-line, and Braniff completed his hat-trick from the subsequent penalty.
Murray added a second a minute later before Ryan McStay got in on the action, and Twigg completed the rout with his hat-trick goal a minute from time.
The result is the biggest victory in local top flight football for almost half a century, equalling Distillery's 11-0 win over Bangor back in 1966.
Elsewhere on Saturday Cliftonville regained the top spot in the league table after a 1-0 victory away to Ballymena United.
Joe Gormley scored the only goal of the game, latching on to Reds keeper Conor Devlin's long clearance upfield before lobbing Dwayne Nelson in the Ballymena goal.
In the game at Dixon Park, Linfield picked up their second win on the bounce with a 3-1 win over Ards.
Aaron Burns opened the scoring for the Blues when he capitalised on Matthew Tipton's superb cross in the sixth minute, but James Cully equalised from the penalty spot in the 16th minute after Billy Joe Burns had fouled Craig McMillan inside the area.
Tipton got himself on the scoresheet when he nodded home a Michael Carville cross midway through the second half, and Burns added his second of the game when he headed in a Carville corner 11 minutes from time.
Also on Saturday Coleraine saw their unbeaten record come to an end as they fell to a 1-0 defeat to Dungannon Swifts at Stangmore Park.
Substitute Chris Lavery scored the only goal of the game in the 73rd minute, reacting quickest when Bannsiders keeper Wayne Drummond could only palm away David McCullough's goalbound effort.
At the Oval, Glentoran saw their time at the Premiership summit cut short by Crusaders, who secured an impressive 3-0 victory over their Belfast rivals.
Colin Coates opened the scoring with a 29th minute header, and nodded in the second from a Chris Morrow free kick in the 56th minute.
Jordan Owens added the third with a great strike, before Coates squandered the chance of a hat-trick when he saw his penalty well saved by Elliot Morris.
Finally Glenvaon survived a scare as they came back from a goal down to earn a 3-1 win over Warrenpoint Town.
The away side took the lead in somewhat controversial fashion when Stephen McCabe dispatched a penalty awarded by the referee's assistant for an infringement inside the area, but Glenavon hit back when Mark Patton headed in the equaliser from Andy McGrory's cross in the 62nd minute.
Guy Bates put the Lurgan Blues in front seven minutes later with close-range finish from Patton's cross, and added a superb second in the final minute when he ran from inside his own half before rounding the keeper to make it 3-1.
Results in full:
Ards 1-3 Linfield
Ards: Cully 15 (PEN).
Linfield: A Burns 6, 79, Tipton 69.
Ballymena Utd 0-1 Cliftonville
Cliftonville: Gormley 3.
Dungannon Swifts 1-0 Coleraine
Dungannon Swifts: Lavery 73.
Glenavon 2-1 Warrenpoint Town
Glenavon: Patton 62, Bates 69, 90.
Warrenpoint Town: McCabe 45 (PEN).
Glentoran 0-3 Crusaders
Crusaders: Coates 29, 56, Owens 62.
Portadown 11-0 Ballinamallard Utd
Portadown: McMahon 7, 74, Twigg 24, 76, 89, Braniff 27, 73, 82 (PEN), Murray 65, 83, McStay 86.