However, the hosts rescued the draw late on as Morata's header saw Julen Lopetegui extend his unbeaten start as Spain manager to nine games.
Despite missing their Real Madrid contingent due to their involvement in the Champions League final last Saturday, Spain were still able to name a strong side with the likes of Gerard Pique, Andres Iniesta and Silva all included in the starting lineup.
Colombia had superstars of their own to call upon too as James Rodriguez, Falcao and Juan Cuadrado - the latter of whom was sent off in that Champions League final - all started, but it was Spain who began the match on top.
Silva was thwarted with an early chance by a good defensive block, but the Man City playmaker had another clear sight of goal in the 10th minute when he was played through by Iniesta only to be denied twice in quick succession by David Ospina.
Koke's effort from the resulting corner flew well over the crossbar after he won possession back on the edge of the box, and Colombia soon offered their first threat of the match when Cuadrado stung the palms of Pepe Reina from outside the area.
Spain still had a good degree of control over the match, though, and Pedro was the next to come close for the home side when he darted inside from the left flank before flashing a powerful strike narrowly over the crossbar.
The hosts did take the lead midway through the first half when Iago Aspas and Pedro combined down the right flank before the latter swung a low ball into the box which Silva got just enough on to squeeze past Ospina, who should have done better.
It was a deserved lead for La Roja, and they almost doubled it on the half-hour mark when Aspas produced an impudent flick from Koke's cross that rippled the side-netting at the near post.
Rodriguez, who has been heavily linked with a move away from Real Madrid this summer, was noticeably quiet during the opening exchanges, but he burst into life with a little over 10 minutes of the first half remaining when he turned smartly on the edge of the area before firing an effort wide of the far post with the outside of his foot.
That sparked a period of improvement from Colombia and Rodriguez had another sight of goal shortly afterwards, but this time Reina made a comfortable save from the in-demand attacker.
In the end, Colombia's equaliser came out of nothing as an aimless ball was allowed to bounce in the box and, while Pique, Cesar Azpilicueta and Reina all left it for each other, Cardona nipped in to lift it over the stranded keeper and beyond the retreating Pique on the line.
That took the two sides into the break all square, but Spain had a chance to regain their advantage within minutes of the restart when half-time subs Marco Asensio and Saul Niguez failed to make the most of a promising opportunity.
Aspas was the next to test Ospina as Spain continued to dominate possession, but it was Colombia - sitting five places above their opponents in the FIFA rankings - who took the lead against the run of play 10 minutes into the second half.
A deep corner saw Falcao rise above Azpilicueta at the back post and power a header past the helpless Reina for his 26th international goal, taking him out on his own as Colombia's highest ever goalscorer.
It also marked the first time since March 2006 that Spain had conceded more than one goal in a home match, and they quickly went in search of an equaliser to retain their unbeaten record under Lopetegui.
Koke tested Ospina with a free kick on the hour mark, but the closest the home side came to restoring parity before the closing stages arrived moments later when Morata flashed a header narrowly wide of the far post after being picked out by the cross of debutant Asier Illarramendi.
It proved to be a valuable sighter for the Real Madrid striker, though, and he produced a pinpoint finish with his next sight of goal to flick a fine header from Saul's delivery across the keeper and into the bottom corner with only three minutes remaining.
The hosts were keen to restart the match quickly and push on for a winner, but a Giovanni Moreno effort which drew a comfortable save from Reina was the closest either side came to another late twist as they settled for a share of the spoils.
Spain will now turn their attention to Sunday's World Cup qualifier against Macedonia in Skopje, while Colombia will stay in Spain for a friendly against Cameroon in Getafe on Tuesday.