Colak found the target on the hour mark to secure a 1-0 win at PSV Eindhoven and seal a 3-2 aggregate victory in their play-off.
The Scottish-born striker tucked the ball away after Malik Tillman had created the chance following a mix-up in the Dutch defence.
Rangers will now go into Thursday's draw having secured a place in the money-spinning group stages for the first time since 2010.
Twelve months ago, Colak was public enemy No 1 for Ibrox fans when he scored twice for Malmo to give the Swedish club victory over Rangers in the third qualifying round.
"It's difficult to describe in words it was a great team performance. It's an amazing journey we'll have this year and I'm so proud of all of these guys," Colak told BT Sport.
"I was in tears on the pitch, it was just a moment of joy. As a kid I always wanted to compete at this level and to be here is amazing."
Rangers will join Glasgow rivals Celtic in the draw, the first time in 14 years that both of the Scottish giants have been in the group stage at the same time.
Australian international goalkeeper Mathew Ryan pulled off a string of impressive saves as Copenhagen resisted a Trabzonspor side looking to overturn a 2-1 first leg deficit.
The second leg ended 0-0.
Dinamo Zagreb made the group stages for the first time in three years with a 4-1 extra-time win over Bodo/Glimt of Norway who had won the first leg 1-0.
Mislav Orsic and Bruno Petkovic put the Croatian side in the ascendancy on the night before Albert Gronbaek levelled the tie on aggregate in the 70th minute.
However, Swiss striker Josep Drmic and Peter Bockaj scored in the dying minutes of extra-time to give Zagreb the 32nd and final place in Thursday's draw.