He initially gave an oral commitment to the University of Minnesota, but after their coach, Tim Brewster, was fired, LaCosse reconsidered and eventually chose Illinois.

[5] Throughout training camp, LaCosse competed for a roster spot as the third tight end on the depth chart against Adrien Robinson, Jerome Cunningham, and Will Tye.

On November 3, 2015, the Giants signed LaCosse to their practice squad after he performed well at a tryout and earned the job over Adrien Robinson.

LaCosse was signed to provide depth after Daniel Fells was placed on injured reserve and Larry Donnell sustained a neck injury.

Head coach Tom Coughlin named LaCosse the third tight end on the depth chart, behind Tye and Jerome Cunningham.

[12] On December 27, 2015, LaCosse made his professional regular season debut and caught three passes for 22-yards during a 49–17 loss at the Minnesota Vikings in Week 16.

[18] Throughout training camp, LaCosse competed against Tye, Jerell Adams, and Rhett Ellison to be the second tight end on the Giants' depth chart.

[19] Head coach Ben McAdoo named LaCosse the fourth tight end on the Giants' depth chart to begin the regular season, behind Rhett Ellison, Evan Engram, and Jerell Adams.

[14] During training camp, LaCosse competed against Austin Traylor, Jeff Heuerman, and Troy Fumagalli for the job as the primary backup tight end.

Head coach Vance Joseph named LaCosse the third tight end on the depth chart to begin the regular season, behind Jeff Heuerman and Jake Butt.

On November 18, 2018, LaCosse earned his first career start and had a nine-yard reception as the Broncos defeated the Los Angeles Chargers 23–22 in Week 11.

[27] With the retirement of Patriots tight end Rob Gronkowski in the offseason, and recently signed Benjamin Watson entering the year under a four-game suspension, LaCosse assumed a significant role in New England's offense.

Following Watson's subsequent release after his suspension, LaCosse officially assumed the starting tight end role in the Bill Belichick- and Josh McDaniels-led offense.