Zirkzee sparked a second-half comeback after going into the break 1-0 down, with Mason Mount later scoring the winner.
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Holland striker Zirkee said afterwards: "Great feeling, it’s always a good feeling but most importantly just got us back into the game and this guy (Mount) scored a pretty good goal after.
"It was all about three points today, just very happy we got them.
"It’s one of the biggest clubs in the world and you have to perform so if you don’t score for a long time it can get to you but I’m surrounded by great players, great people and we help each other every day.
"So they made it quite easy for me and today is just a reward I think for patience, hard work, trying to be consistent and I’m just thankful to them."