Mackay was shown a straight red card after referee John Beaton ruled he had brought down Derk Boerrigter inside the area - Nir Biton converted the 75th-minute spot-kick to put Celtic two up.
Boerrigter went down after a delay after Mackay had brushed into him as the Dutch winger ran into the box.
A statement on St Johnstone's official twitter feed read: "We can officially confirm that we will be lodging an appeal against Dave Mackay's red card received during last night's game."
Mackay had claimed for a penalty minutes earlier after a challenge by Celtic defender Virgil van Dijk in the other area.
The decision means Mackay will be free to play for Tommy Wright's side against Hamilton in the Scottish Premiership on Saturday but will miss the visit of Aberdeen the following week if his appeal fails.