The visitors had looked set to claim the win after Edwin Cardona and Radamel Falcao put them ahead following David Silva's opener, only for Morata's brilliant header to rescue a draw for the home side.
The goal spared head coach Julen Lopetegui from his first defeat since succeeding Vicente del Bosque last year, although it marked the first time that Spain have conceded two goals in a home match since a 3-2 win over Ivory Coast in 2006.
A minute's reflection was held prior to kick-off to mark the deaths of former Colombia international Marcos Coll, the ex-president of the coaching committee in Murcia, Manuel Redon, and Ignacio Echeverria, a Spanish citizen killed in the terror attack in London last Saturday.
Lopetegui named a strong starting line-up that included veterans Gerard Pique, Andres Iniesta and Silva alongside debutant Asier Illarramendi, and Spain deserved the lead they took when the Manchester City star netted after 22 minutes.
A dreadful defensive mix-up allowed Cardona to score a well-taken equaliser, however, before Falcao's bullet header shortly after half-time – his 26th international goal, two years on from his 25th – left Estadio Nueva Condomina stunned.
Spain enjoyed 71 per cent of the ball but looked unlikely to stop a resurgent Colombia from claiming a third win in a row, until they were saved by an expert finish from Real Madrid man Morata, who continues to be heavily linked with Manchester United.
2 - Spain have conceded two goals in a game at home for the first time since March 2006 (3-2 against Ivory Coast). Surprise. pic.twitter.com/c9CRcL5e2m— OptaJose (@OptaJose) June 7, 2017
Silva had the first good opportunity after eight minutes, expertly controlling Iniesta's throughball before seeing a shot saved by David Ospina, a rebound blocked by Carlos Sanchez and a third attempt held by the goalkeeper.
Juan Cuadrado tested Pepe Reina with a powerful strike from 25 yards but Spain continued to look the more threatening, with Pedro curling a fierce left-foot effort over the bar after a fine run.
With 21 minutes played, they made their dominance count. Koke fed Pedro down the right and the Chelsea man's low cross was side-footed goalwards by Silva, with Ospina just unable to stop the ball bouncing into the net.
Iago Aspas and James Rodriguez each came close with clever efforts but it was Cardona who levelled the scores before the break after making the most of some desperate Spain defending.
Pique – who had been booed by some of the home support once again – and Cesar Azpilicueta both hesitated to deal with a high ball into the box from Cuadrado, allowing Cardona to steal in and loop a clever finish over the stranded Reina.
Silva and Iniesta picked up knocks in the first half and both were withdrawn at the break, with one of those replacements, Real Madrid's Marco Asensio, almost setting up Aspas to regain Spain's lead less than 10 minutes after the restart.
However, it was Colombia who struck next in the 55th minute, with Falcao leaping highest to power home James' corner after Reina found himself stranded out of his goal.
Koke's free-kick required an athletic stop from Ospina as Spain went in search of an equaliser, but even with Morata on the pitch, they fashioned few other chances of note.
However, with three minutes of normal time remaining, Morata met Saul Niguez's cross from the left with a superb glancing header that flew into the corner beyond Ospina and gave Spain something of a lift ahead of next week's World Cup qualifier against Macedonia.