Cliftonville enjoyed a dream start when Ballymena defender Daithi McCallion underhit his pass back to Bradley Wade, allowing Robertson to nip in ahead of the goalkeeper to slot home the opener.
After sustaining a short spell of pressure, the Reds doubled their advantage as Josh Kelly exchanged passes with Rory Hale before driving into the box and squaring for Healy, who shrugged off Stephen O’Donnell and coolly slotted beyond Wade.
Moments later, Jim Magilton’s men threatened again when Hale slid a neat through ball into Robertson, who chopped inside and drove towards the near post, but Wade spread himself well to make the save.
However, after the restart, the contest turned on its head when Hale was shown a straight red card for a late challenge on Rutkowski in the area.
The away side took full advantage as Rutkowski stepped up to take the resulting spot kick himself and made no mistake from 12 yards.
After defending for large parts of the second half, Cliftonville almost restored their two-goal cushion when Kelly launched a long throw into the area, which Michael Glynn diverted into the path of Healy, whose backward header clipped the upright.
Midway through the second half, the turnaround was complete when substitute James Hood’s low effort was deflected beyond Morrison by Joe Toole to draw the visitors level.
Ballymena dominated the remainder of the game and continued to probe, but were unable to find a late winner.