The Croatia striker has endured an injury-plagued 18 months but he brushed aside recent concerns about his fitness by scoring in the 24th and 40th minutes.
Nicklas Bendtner, who earlier dismissed reports linking him with a move to AC Milan, also struck twice before the break (17, 43) to put the Gunners firmly in control.
After half-time it was the turn of Robin van Persie as the Dutchman converted a 66th-minute penalty when Bendtner was fouled. He thought he had added a sixth with a header, but it was ruled out.
Szombathelyi had the chance to pull one back in the 83rd minute when William Gallas was adjudged to have felled Matel Skriba in the box, but Barbosa slipped in his run-up and sent his spot-kick past the post.
David Nugent grabbed a hat-trick for Portsmouth as they cruised to a 6-1 victory over Eastleigh at the Silverlake Stadium.
The non-league side were put to the sword with Younes Kaboul, Paris Cowan-Hall and Danijel Subotic also on target for Paul Hart's side.
Leeds heldDerby County boss Nigel Clough saw a young Rams side claim a narrow 3-2 success away to Matlock Town with Alex Prijovic finding the net 10 minutes from time.
Arnaud Mendy and Greg Mills were the other men on target for the visitors, with the home side twice restoring parity at the Causeway Stadium before Prijovic's winner.
Leeds United had to settle for a 3-3 draw against Bradford Park Avenue as the UniBond League side hit back from 2-0 and 3-1 down.
Enoch Showunmi and Tom Elliott put the Whites in control early on, only for Chris Hall to pull one back shortly before the break at Horsfall Stadium.
Alan Sheehan restored Leeds' cushion from the penalty spot after Liam Darville was fouled, but further goals from Mark Bett and Hall left honours even.
Dagenham & Redbridge left it late to claim a 2-1 success away to Braintree Town with Tommy Tejan-Sie finding the net in the 89th minute.
Wesley Thomas' shot from the edge of the box in the 14th minute put John Still's men ahead, only for their hosts to level with time running out.