The home side had to weather the storm in the closing stages, but over the 90 minutes were worth the victory.
Inverness made a bright start and won a couple of early corners but they came to nothing and Hearts made them pay.
With just eight minutes gone, Audrius Ksanavicius slid the ball through to Mikoliunas, who calmly took the ball past Michael Fraser before slotting the ball home.
It was the start that Hearts had wanted after the previous week's defeat at Falkirk and it gave them the confidence to take the game to Inverness.
Lee Wallace then sent Michael Stewart clear with a long ball out of defence, but the midfield player's shot was touched over the top by an alert Fraser.
Laryea Kingston then tried his luck after eluding a challenge from Dougie Imrie but his shot was just too high.
Stewart was pulling the strings in midfield and Fraser had to look lively to intercept a through ball for Jamie Mole.
There was a scare for Hearts when Don Cowie whipped the ball across goal but there were no takers.
Inverness took heart from that and Marian Kello made a meal of dealing with an Ian Black shot, but Imrie could not take advantage and fired the loose ball straight at the goalkeeper.
Mikoliunas responded for Hearts and tried his luck from distance shortly before half-time, but the ball was always going over and then Kingston sent a header onto the roof of the net.
Mole looked set to add to Hearts' lead early in the second half, but Grant Munro somehow managed to get across to make the block.
Adam Rooney had a chance to equalise shortly afterwards, but Marius Zaliukas was alert to the danger and managed to get his body in the way of the shot.
Kello then made a fine save to deny Roy McBain after some slack play at the back from Hearts.
Kingston had a late effort for Hearts but Fraser saved comfortably.
Caley Thistle finished strongly, but Hearts held firm at the back to hold onto the three points.