The Portman Road side snatched the lead in dramatic fashion in the fourth minute when striker Jonathan Walters played the ball into the path of Quinn, who drilled a shot from 25 yards into the top corner.
Wednesday, desperate to climb clear of the relegation pack, were back on level terms seven minutes later after Jermaine Johnson shrugged off a challenge from Dan Harding and his stinging shot was deflected by centre-back Jason De Vos onto the bar.
Burton O'Brien's corner was cleared straight to Marcus Tudgay, who hammered the ball past new signing Stephen Bywater from the edge of the area for his sixth goal of the season.
Ipswich had a chance to snatch back the lead as Quinn pumped the ball to the incoming David Norris, but he sliced his header wide of the target from close range.
Tudgay was handed a half-chance as he beat his marker to O'Brien's corner but his header lacked power as Bywater snatched the ball.
Wednesday keeper Lee Grant foiled midfielder David Wright by diving to his left to snatch his close-range strike, before Ipswich grabbed their second in the 71st minute.
Centre back Alex Bruce's deep cross found Quinn and his downward header reached substitute Alan Lee, who with his back to the goal, hooked the ball over the line.
Grant was lucky to stay on the pitch minutes later when he failed to snatch the ball and, as Norris closed in he appeared to bundle the midfielder down as he tried to stop him shooting.
However, Ipswich's appeals for a penalty were brushed aside by referee Colin Webster.
O'Brien went close with a fierce strike which was stopped in spectacular fashion by Bywater as Wednesday staged a late rally in an effort to salvage a point.