Heads were turned before the season kicked off last week after Toronto managed to secure the services of Giovinco and Altidore from Europe.
United States international Altidore netted a brace in his first competitive outing last week as Toronto accounted for Canadian rivals Vancouver Whitecaps 3-1, while Italian recruit Giovinco - the league's highest paid player - set up the former's opener.
But the duo failed to find the back of the net as the Crew made their numerical advantage count in the second half at Columbus Crew Stadium.
Toronto defender Justin Morrow was controversially sent off on the stroke of half-time after bringing down Ethan Finlay just outside the penalty area and the home team took advantage following the break with goals to Justin Meram and Kei Kamara for the Crew's first win of the campaign.
Meram headed home the opener in the 57th minute via a cross from Finlay before Kamara doubled Columbus' lead four minutes later after striking past Toronto goalkeeper Joseph Bendik.
There were goals galore at Rio Tinto Stadium in Utah as Alvaro Saborio's 86th-minute penalty earned Real Salt Lake a 3-3 draw with Philadelphia Union.
RSL trailed 2-1 at the break after a double from Fernando Aristeguieta cancelled out Javier Morales' opener.
The hosts levelled proceedings through Jamison Olave 10 minutes into the second half, only for the Colombian centre-back to put the pull into his own net three minutes later.
However, Saborio spared Olave's blushes from the spot after Luke Mulholland was fouled in the area with four minutes remaining.
Dallas top the Western Conference with a perfect record from two matches following their 3-1 win over Sporting Kansas City.
Blas Perez inspired Dallas thanks to his two-goal haul.
In other results, Chris Wondolowski's brace lifted San Jose Earthquakes past Seattle Sounders 3-2, while Octavio Rivero's 86th-minute goal gave the Whitecaps a 1-0 win at Chicago Fire.
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