The Scottish champions frustrated the Italian club for long spells in a near-empty stadium following Inter's crowd ban by UEFA.
But, for all their endeavour, Alex McLeish's side could not create much and Inter could have won by a greater margin after missing a first-half penalty.
It was left to Pizarro to inflict the damage with a deflected free-kick early in the second half.
If it was hard on the Scottish side, at least they emerged with some credit after their dismal domestic form.
Rangers went straight on the attack from the kick-off and Barry Ferguson signalled their intent with a shot after only 20 seconds, but it was always going too high.
Obafemi Martins had the ball in the net for Inter shortly afterwards but it was ruled offside, when television evidence later showed the strike should have counted.
Santiago Solari then tested Ronald Waterreus with a low shot but it was Rangers who created the best early chance after ten minutes.
Thomas Buffel whipped in a great cross from the right and Peter Lovenkrands volleyed narrowly over from six yards.
Rangers defended competently to frustrate Inter in the first half and created a couple more chances of their own.
Buffel sliced well wide after a slick Rangers move, as Inter looked to have run out of ideas up front.
But Rangers had a major let-off eight minutes from half-time when Waterreus upended striker Julio Cruz after Olivier Bernard had earlier given away possession.
The goalkeeper was booked rather than red-carded as Cruz was veering away from goal when he was pulled down.
The Argentinian took the penalty himself and sent Waterreus the wrong way but the ball struck the post.
Moments later, Ian Murray gave away possession and Martins was unlucky to see his shot brush the side-netting. As half-time approached, Luis Figo saw a downwards header bounce over the crossbar.
Inter finally made the breakthrough four minutes after half-time when a deflected free-kick from Pizarro left Waterreus with no chance.
Rangers tried to get back and Dado Prso just failed to get on the end of a head-flick from Hamed Namouchi in front of goal.
But Solari somehow missed in front of an empty net in the 73rd minute after a quickly-taken free-kick had caught out Rangers.
The Gers did not have much to offer in response and Inter held on for the win.