Ipswich should have been in front after Shefki Kuqi headed in after Pablo Couñago's shot had rebounded off a post but the goal was ruled out for offside, even though television cameras proved otherwise.
He had already threatened before he put Barcelona ahead after 13 minutes, reacting quickly to fire a low, angled finish past Hart after Messi's cross had rebounded off Kompany.
In a scrappy first-half, Gabon thought they had taken the lead after 16 minutes when Romuald Ntsitsigui reacted quickest to bundle the ball over the line after it had rebounded off a post only for the linesman's flag to go up.
They twice came close to the game's first points, but Marc Sneyd was unable to gather cleanly near the try line after his kick had rebounded off Wigan full-back Matty Bowen, then Luke Dorn's effort was controversially ruled out for obstruction by video referee Ben Thaler.
"I'm not sure how it happened – it must have rebounded off the table and it hit me on the leg.
The goalkeeper rightly tried to avoid handling the ball from the deliberate back pass – he only intervened with his hand in a second phase of play, after the ball had rebounded off the post.
The onslaught continued in the second half as substitute Dowie nodded home after her original header from Carney's cross had rebounded off the bar.
Mane, 22, struck from 10 yards just past the half-hour mark, thumping the ball in after Graziano Pelle's low drive had rebounded off the post.
The hosts raged when referee John Beaton disallowed it a good 10 seconds after Sutton curled the ball into the top corner, but it had only come to the big Englishman because an Aberdeen clearance had rebounded off Motherwell's Stuart Carswell.
First, Ronaldo, following more good work from Bale, ran 30 yards before drilling in before Morata swept in Bale's pass, after Ronaldo's shot had rebounded off the bar.
