The draw sends the Group E winners to Austria on February 16 or 17 with the return leg taking place at the Riverside Stadium a week later.
Graz lost their Champions League qualifier against Liverpool 2-1 on aggregate despite winning 1-0 at Anfield and were defeated 3-0 by Rangers at Ibrox in Group F but still progressed.
McClaren was happy enough with the draw, but admitted he would be hoping for a little help from his Barclays Premiership peers.
“We will,” he said. “Don Mackay (European scout) will be getting together the videos and the reports, so we will be well briefed.
“It is the last 32 and no game would have been easy, but it is a good draw for us. They are the Austrian champions, so they have a good pedigree.
“They played Liverpool in the Champions League qualifiers. They lost 2-0 at home, but won 1-0 at Anfield, so they are no mugs and it will be a test for us.
“I am certainly delighted we are not heading somewhere like the Ukraine or Russia. This is a good opportunity for us.
“We are delighted with the draw, and coming back to the Riverside for the second leg, we will know what we have to do.
“It is their winter break now and our game with them will be their first one back. I am not sure whether that will be a hindrance or a help.
“I suppose it will hinder us in our preparations because we will not see them play before we actually play them. They may have a couple of friendlies, but they will be hard to gauge.
“Then again, it may be an advantage because they will have had a period where they have not played. But either way, it is going to be a difficult first leg.”
The away leg will be played at the Arnold Schwarzenegger Stadium, a venue McClaren has encountered previously with both Manchester United and England.
“It is a nice little stadium, tight, probably something like in Ostrava when we played them, so we are used to that,” he said. “It will be interesting.”
If Boro manage to get past Graz, they will face either Sporting Lisbon, who finished third in derby rivals Newcastle United’s group, or Feyenoord, although McClaren is not looking that far ahead.
“We are not looking at that tie at the present moment,” he said. “We need to make sure we get through this one first. We cannot look beyond it.”
Newcastle’s reward for topping Group D was a clash with Heerenveen, the club from which they signed striker Jon Dahl Tomasson for £2.2million in July 1997.
And the Magpies have enjoyed success over Dutch sides in recent seasons.
They claimed a memorable 3-2 victory over Feyenoord in Rotterdam to clinch their place in the second group stage of the Champions League two seasons ago and defeating both NAC Breda and PSV Eindhoven on their way to the UEFA Cup semi-final during the last campaign.
Skipper Alan Shearer was happy enough with the draw, which pitches the winners into battle with either Olympiacos or Sochaux, the side the Magpies beat 4-0 in France last month, in the following round.
“It’s not a bad draw for us,” he said. “We could have got a lot worse. We are not having to travel too far. We are not playing a side from Eastern Europe as we could have done if we had come second in the group.
“The best thing about it is that we will have tremendous support there. It is not too far to travel and the lads will be happy that it is not too far to come back to prepare for the next Premiership game.
“The other good thing is that we are playing a Dutch side, so you can guarantee that we will be playing a side which will play football.”
For all the latest UEFA Cup results and fixtures, click here.
UEFA CUP DRAW
Funny, before the match against Boro Benitez gave advice to McClaren about Boro's Spanish opposition. A week later, Boro ended up losing 2-0
