Goren had a goal and an assist to power the Moose past the Marlies 4-1 before an announced crowd of 1,869 spectators at Ricoh Coliseum - the smallest gathering in the Toronto team's history.
Toronto is fifth in the North Division, eight points behind Grand Rapids for the final playoff spot. But the Griffins have three games in hand on the Marlies, who have just 25 games left in the season.
"The urgency has crept in for the last little while," said Toronto defenceman Jay Harrison said. "I think we know how difficult it is to make points up (in the standings) this time of year.
"We can't keep giving up points at home."
Brad Moran, J.J. Hunter and Nathan Smith also scored for Manitoba (30-16-7-4), which extended its season-high win streak to eight games.
Robbie Earl replied for the Marlies (22-29-2-4), who've lost three of their last four and are a league-worst 8-17-2-1 at home.
Playing before an announced crowd of 1,869 spectators at Ricoh Coliseum - the smallest gathering in team history - the Marlies took a 1-0 lead at 16:23 of the second. Earl broke in from Manitoba's blue-line and fired a shot over the shoulder of Moose goalie Wade Flaherty.
Moran fought his way through a scramble in front of Toronto's and scored with less than four seconds remaining in the period.
"(Moran) put it in (sitting on) his butt," said Marlies goalie Justin Pogge, his club's best player in the loss. "It was a bad-luck bounce; not much you can do."
Harrison admitted the late goal killed whatever momentum the Marlies had built up against one of the league's hottest teams.
"It was a bit of a heart-breaking goal," he said. "Any goal in the last minute of a period, you look to avoid those and really bear down."
The Moose went ahead for good at 10:18 of the third on Smith's 11th goal of the season. Goren made it 3-1 at 14:18 with his 17th goal before Hunter scored into an empty net at 19:46.
Flaherty (13-9-2), made 31 saves for the Moose while Pogge (10-16-1) stopped 25 shots.