Rookie running back Will Ford, a recent addition to the Bomber roster, scored the winning touchdown from five yards out with less than two minutes left.
The Bombers, now 2-5, started slow, putting only a single point on the board in the first quarter.
It was only Elliott's third start at quarterback since he put on a Bomber uniform three seasons ago.
And it was veteran Hamilton quarterback Henry Burris who took command early, scoring a touchdown on his first drive with a 37-yard pass to Chris Williams.
He wanted the Ticats to take command in the East with a win that would have lifted them above Toronto and Montreal but instead they fell to 3-4.
Burris personally handed the ball to the Bombers three times on fumbles that either killed Hamilton's momentum or led to Winnipeg scoring plays during the game. The TiCats had three other turnovers as well.
The Bombers only point in the first quarter came on a single from a missed 49-yard field goal by Justin Palardy.
Williams scored a second touchdown in the second quarter on a punt return but Palardy kept the Bombers in the game with two field goals to end it 14-10 for Hamilton. He added another in the third.
Hamilton's Luca Congi had one field goal, a 43-yarder in the third.
In the third, Elliott led the Bombers on a four-play 78-yard drive to their first touchdown with a handoff to Chad Simpson, tying the game at 17-17.
Winnipeg finally took the lead in the fourth when Elliott connected with Chris Matthews for the Bombers' second touchdown and it was 25-17 after the convert.
But Burris came back with an 84-yard pass to Williams for his third touchdown of the night then Aaron Kelly added a two-point convert to tie the game again.
The Bombers replied on the next drive with Ford's touchdown, after Simpson was injured and left the field.
It was the first home game of the season that wasn't a sellout for the Bombers, although more than 27,000 showed up.