Ciara Grant put Ireland ahead on 59 minutes with a penalty after Sonya Hughes's free-kick was handled in the box.
However, the visitors hit back to equalise five minutes later when the unmarked Kozhnikova swooped low to steer a header from Morozov's free-kick past Emma Byrne and into the corner.
Russia, who had lost just once in their six previous group matches, were the livelier in the opening exchanges and Ireland goalkeeper Emma Byrne was on alert after just three minutes when Oksana Shmachkova cut in from the left and tested the Arsenal stopper with a shot.
Two minutes later, Elena Danilova took advantage of hesitant defending to hook a right-foot just a yard beyond the far post while Ekaterina Sochneva brought the best out of Byrne with a free-kick from distance on 13 minutes.
The first threat Ireland made on the visitors goal was two minutes later as Fiona O'Sullivan raced onto a long ball but the full-back managed to poke the ball back to her ‘keeper at the crucial second.
Pressure remained on the Russian goal thereafter, especially when Aine O'Gorman skipped past her marker on the half hour and had her tempting low cross cleared for a corner.
It was the hosts that continued to hold the upperhand following the turnaround with O'Sullivan not far off with a shot on the turn inside the box.
However, when the striker was fouled on the edge of the box with 58 minutes played, set-piece specialist Sonya Hughes blasted her effort high into the wall where Alla Rogova was adjudged to have handled. Grant's resultant spot-kick was fired down the middle past the stranded Elvira Todua and into the net.
Ireland barely had time to consolidate when they were pegged back. Morozov swirled in a free-kick from the left side which caught the Irish defence unawares, allowing the unchallenged Kozhnikova to nod beyond past Emma Byrne from eight yards.
Byrne was kept busy during the ensuing minutes as Russia pushed for a second. Tatiana Skotnikova's thunderous strike required the Irish netminder to scamper across and tip away for a corner.
Substitute Kurochkina stabbed a volley just wide with 20 minutes remaining and, at the opposite end, the girls in green almost claimed the three points five minutes from time
when Russia's goalkeeper Todua dropped a cross from O'Gorman but Shmachkova reacted well to sweep the ball off the line as Meabh De Burca was poised to pounce.
Ireland will complete their campaign with another home game on Wednesday when Israel visit the Carlisle Grounds in Bray (7.30pm).
Republic of Ireland: Byrne, MT McDonnell, Fahey, Tracy, S.McDonnell, De Burca, C.Grant, Hughes, Smyth (Russell, 73), O'Gormand, O'Sullivan (Roche, 81).
Source - fai.ie
