Luke Varney fired against the woodwork as Wednesday looked to fight back, but the visitors moved further ahead with 23 minutes remaining as the impressive McAnuff played a neat one-two with Marek Matejovsky before laying on a simple finish for Kalifa Cisse.
Wednesday drop below Reading in the table as a result of their defeat and have now lost four consecutive matches on home soil for the first time in more than nine years.
Reading can also take heart from a first clean sheet in 13 games.
Owls boss Brian Laws made three changes to the side that were thumped 4-0 by leaders West Brom last weekend, with Mark Beevers, Etienne Esajas and Varney coming in for Frank Simek, Jermaine Johnson and Leon Clarke.
Cisse replaced the injured Brian Howard in Reading's starting line-up, while Rasiak was preferred to Jimmy Kebe in the only other change from last weekend's 2-1 defeat at Derby.
Neither side was at their best in a dismal first half which contained little in the way of goalmouth action.
Reading were forced into a change after 23 minutes as Simon Church picked up an injury.
Kebe came on in his place and moments later Owls midfielder James O'Connor scuffed a shot wide from 20 yards.
Gylfi Sigurdsson had Reading's first effort of note in the 31st minute, but the Icelander's shot sailed harmlessly over the bar.
Sigurdsson came closer five minutes later when he curled a free-kick inches wide and the Royals threatened again before the break as Rasiak weaved past two men before being denied by Wednesday goalkeeper Lee Grant, who bravely claimed the ball at the Pole's feet.
Mercifully, things improved after the interval and Rasiak saw two further opportunities go begging before opening the scoring with a well-placed header from McAnuff's cross.
Tommy Spurr then fired wide for the hosts before Rasiak saw another header saved by Grant.
With chances now coming thick and fast, Wednesday's Marcus Tudgay saw a 20-yarder flash wide of Adam Federici's left-hand post before Varney struck the post with a volley from inside the area.
However, it was Reading who grabbed the game's second goal as a slick passing move culminated in Cisse stroking home from six yards.
Wednesday threw men forward in the closing stages, with Johnson, Tudgay and Clarke all missing chances to reduce the deficit.
Clarke also saw a late strike ruled out for offside as Reading held on for a valuable three points.