Second-half substitutes John Cofie and Ryan Tunnicliffe - both of whom featured in the Under-18s' draw at Sheffield United on Saturday - then completed a resounding 5-1 win.
Eikrem engineered the subs' strikes with a couple of sublime passes. For United's fourth goal on 76 minutes, he played a reverse pass into Nick Ajose's path inside the area and when the striker pulled it back, Cofie tucked home first time. For the final effort, nine minutes from time, Eikrem was the wall in a one-two - dinking the ball back to Tunnicliffe who needed just one cool touch to beat the advancing Branagan.
The night was not without its concerns for Solskjaer and his assistant Warren Joyce, with injuries enforcing the withdrawals of Oliver Gill before kick-off, Ravel Morrison in the first half and Vermijl in the second period. And although some slack possession at times threatened to offer Bury a way back into the game, the overall display was very positive and bodes well for Thursday's tie in the Manchester Senior Cup, at home to Bolton.
United: Johnstone; Vermijl (Cofie, 63), Wootton, R Brown, Fryers; C Evans (Tunnicliffe, 63), Eikrem, Morrison (M Keane); Stewart, Ajose, Brady. Sub not used: Devlin.