The Scottish champions dominated the first half, but a rejuvenated Villarreal team went on the attack in the second and did enough to secure their quarter-final berth.
Rangers showed their intent early as Chris Burke and Hamed Namouchi both made long-range attempts on goal which went well wide.
But the visitors made the breakthrough on 11 minutes when the ball broke to Lovenkrands after good work from Barry Ferguson and Thomas Buffel.
With Villarreal keeper Sebastien Viera off his line the Dane had a clear shot on goal but did well to keep his composure and find the back of the net to make it 1-0.
Rangers stayed in control of the first half but began to come under pressure from the Spaniards as half-time approached.
On 35 minutes Juan Roman Riquelme's well-positioned free-kick found Diego Forlan in the box but he couldn't control his header.
Five minutes later, Rangers were caught on the break by Forlan, who raced the length of the pitch before seeing his effort blocked by the ever-steady Ronald Waterreus.
Villarreal made two changes during the interval and that proved to be the spark they needed as they wrenched control of the match from the Scots immediately from the restart.
Rodolfo Arruabarrena equalised for Villarreal less than four minutes into the second half picking up a cross from Forlan and slotting the ball past Waterreus.
Determined to win back their lead Rangers threw everything forward but were caught on the break by Villarreal with substitute Guillermo Franco forcing two good saves from Waterreus.
Kris Boyd joined the fray midway through the half and could have got the vital goal for the visitors, but he mistimed his effort from a Chris Burke cross.