It's not the start that the new manager would have wanted at home.
The visitors looked more dangerous in the first-half, getting down both flanks quite well.
Darren Randolph had to be alert to keep an effort out from Pukki on nine minutes, after the Irish line failed to hold its shape.
Ireland started the second-half brightly but fell behind to a Jensen goal, and the Finns looked more likely to double their advantage than to concede an equaliser.
The possession stats were the only positive, and there is plenty for Stephen Kenny to think about ahead of the big one against Slovakia in just under five weeks' time.
Darren Randolph - 8
Made himself big to deny Pukki in the ninth minute. Saved another save on 21 minutes, again from Pukki, but it was more difficult for the Norwich forward than the first chance.
Could do little for the goal, and was alert on 72 minutes to keep another Pukki effort out. Stood up strong to keep a Taylor effort out five minutes later.
Tipped an effort over the bar from Alho in the last five minutes as the visitors finished strongly.
Enda Stevens - 5
Caught a couple of times with through balls early on. Some loose passes at the crucial time when in attack in the first-half.
Put in a really poor cross on 51 minutes under no pressure to throw away a good attacking opportunity.
Shane Duffy - 6
Cut out a through ball on 22 minutes to concede a corner. Read the play well as the Finns broke on 52 minutes as he cut out a back heel in his own box.
Headed a late chance over from a corner.
John Egan - 7
A composed performance from the Cork man.
A good tackle to concede a corner, but prevent a clear chance for Finland in the third minute. Got on the end of Brady's free-kick on 39 mins, but his effort was blocked for a corner.
Did well to get across and prevent Pukki getting in for a one on one with 15 minutes to go.
Matt Doherty - 6
A solid shift - split the Finnish defence with a lovely pass to set up Robinson in the second-half.
Robbie Brady - 6
Spread the ball well early on. Lost control at the crucial point after a powerful run on 37 mins.
Curled in a lovely cross on 51 mins from a free-kick, but no Irish head go to it. Put the Finnish keeper under pressure to win the ball and set up a chance for Connolly on 57 mins.
Harry Arter - 6
Stung the keeper's hands with a shot on 13 minutes. Saw another effort take a nick as it went just to the wrong side of the Finnish right post in the last five minutes.
Jayson Molumby - 6
Was under a bit of pressure early on in his senior international debut, before winning Ireland’s first corner on 20 minutes.
Booked on 65 mins for a slide tackle after losing the ball on the edge of the finish box. Was unlucky to push an effort, which he hit first time, wide with 10 minutes to go.
Callum O'Dowda - 6
Looked dangerous early on, including an inviting cross from the right on 13 minutes. Rushed a cross on 17 minutes, smashing it high and straight over the end line.
Got forward well on 56 minutes, but his cross was partially blocked and landed at the feet of their goalkeeper.
Tried to curl one into the left corner on 58 minutes, but tweaked an injury in doing so, and it was his last act.
Aaron Connolly - 7
Did well to break up a dangerous looking Finnish attack on 19 mins. Won a free-kick on 39 mins which gave Ireland a good platform.
Drew another free in a dangerous position just after the break.
Had a chance on 57 minutes, but took it slightly wide and narrowed his own angle, before shooting into the side netting.
Couldn’t connect with a near perfect cross from Robinson on 67 mins. Worked hard moments later to work a shot from the edge of the box, but it was always easy for Hradecky to hold.
Adam Idah - 7
Tried hard to get on the ball early on. A few crosses into the box, but didn’t find an Irish body. Did really well on 57 minutes to set up Connolly for a chance.
Worked hard for his near 70 minutes on the field.
Callum Robinson (replaced O’Dowda on 60 mins) - 7
Put in a beautiful cross on 68 minutes but Connolly and McGoldrick didn’t connect. Did well moments later to force a Finnish mistake, which set up McGoldrick with a chance.
Had a big chance to equalise on 70 minutes but was stopped by Hradecky.
David McGoldrick (replaced Idah on 66 mins) - 6
Might have put off Connolly for a chance on 67 minutes. Then hit the ball straight at the Finnish keeper after being set up by Robinson moments later.
Tried a cross field pass with Ireland forward in numbers, and found a Finnish player.
Tried his luck from the edge of the box in stoppage time, with arguably better options on either side.
James McClean (replaced Connolly on 76 mins) - 6
Was lively on the left wing, but his few crosses didn’t give Ireland attackers the chance to get on the end of them.