The Scottish champions sprang a surprise by naming Alexander in goal, with first choice goalkeeper Allan McGregor on the bench, and boss Walter Smith no doubt mindful of the Co-operative Insurance Cup final looming this month.
And the move paid off as Rangers claimed a goalless draw that will give them hope of finishing the job in front of their own fans when the Dutch travel to Ibrox for next week's second leg.
El-Hadji Diouf again took on the lone striker role he performed in the last round against Sporting Lisbon and Rangers were also boosted by the return of Kyle Lafferty after he recovered from tonsillitis.
Young midfielder Kyle Hutton, who yesterday signed a new deal with the Scottish champions, was also handed a starting place for the clash, having already featured in the Champions League this season against Manchester United and Bursaspor.
There was drama for Rangers before a ball was even kicked with the news assistant manager Ally McCoist had been handed a two-match touchline ban following his altercation with Celtic boss Neil Lennon after last week's Old Firm derby.
Once the game was under way, PSV were out of the traps quickly and Marcus Berg went to ground in the box under pressure from Madjid Bougherra but there appeared to be little in the incident and there was no penalty award.
Berg could have grabbed the opener when he tried to run onto a slick pass from Ola Toivonen but Alexander raced off his line to make a brave block before Berg could connect and slot home.
The home side had the best of the chances early on and Jeremain Lens was next to have a crack at goal but his shot was blocked by Kyle Bartley as PSV continued to press for the opener.
Lens was threatening again, this time weaving his way past a couple of Rangers players only to screw his shot over the crossbar.
PSV then passed up a great chance to take the lead when Balazs Dzsudzsak's free-kick was flicked on by Lens for Marcelo who somehow managed to nod wide of target when he looked certain to leave the net bulging.
Rangers looked more lively after the restart and Richard Foster embarked on an impressive run that took him into the box but his attempt to pick out Diouf in front of goal was poor and the danger was cleared.
PSV still looked dangerous and had the Rangers defence under pressure when Toivonen's pass was collected by Berg only for the forward to drag his shot agonisingly wide of the upright from close range.
Stanislav Manolev was fortunate to escape a caution when he appeared to go down easily in the area under pressure from Bartley.
The on-loan Arsenal defender then almost claimed a crucial away goal for Rangers when he threw himself in front of Diouf's corner and connected with a firm header that was touched wide by goalkeeper Andreas Isaksson.
Rangers made their first change of the night after 64 minutes when Diouf was withdrawn for Vladimir Weiss, who had been rated doubtful with a bruised foot.
Marcelo had already squandered a decent chance in the first half and he was left holding his head in his hands again after nodding narrowly past the post when he should have at least found the target.
Lafferty then hooked wide for Rangers, before making way for the introduction of youngster Gregg Wylde for the last 10 minutes.
Smith's men were able to see out the remainder of the game and return to Glasgow with a clean-sheet ahead of the second leg.