The United coach watched on miserably as his side unravelled in the second-half against his former club and eventually conceded a cheap goal after a passage of poor team defending.
Twente scored a dream goal upon their return to European competition after more than a decade away when full-back Bart van Rooij went on a monstrous run past four United players on his way to the edge of the box.
Manuel Ugarte later slid in to clear the impending danger only for Christian Eriksen - who opened the scoring with a stunning strike - was cheaply dispossessed giving Sam Lammers a clear run at Andre Onana's goal before finishing with a powerful effort.
It was another game of big misses for United as Marcus Rashford, Amad Diallo, Bruno Fernandes and Joshua Zirkzee all impressed in build-up play but each failing to score against the Eredivisie side.
Erik ten Hag vented his frustration at his Manchester United players in Wednesday's Europa League draw
Ten Hag later turned to substitutes Alejandro Garnacho and Rasmus Hojlund, but neither were able to affect the scoreline as the hosts were held to an unlikely draw.
Fuming after the match, Ten Hag told TNT Sports: 'We kept them alive when 1-0 up controlling the game and then you have to be consistent and keep going in the second half, we dropped the level and gave a goal away.
'We didn't finish it off, we had to go for the second goal and they stay alive and we get punished with a mistake from ourselves.
'The goal it's a team, the player of Twente can dribble over the whole pitch without being stopped but then can't give a goal away like this.
'It's about we have seen some good structure but you have to keep going for the whole game and when 1-0 up you have to keep going for the second. The first game is important you get a win.'
Meanwhile, United midfielder Eriksen also slammed his team-mates for not wanting to win more than Twente's players and Ten Hag admitted his players dropped below the standards required.
He later added: 'It was the game of their life, they fought for every yard and we didn't. 99 per cent is not enough, you have to give 100 per cent, you have to kill the game, finish it off, we are leading 1-0, but you have to go for the second goal, then you kill the game.
'Especially as we are very ambitious and when you have ambition you have to perform, prove the point and especially today in the second half we were too complacent, we didn't bring it over the line and, as a team, you have to bring this.
'I think often the mentality is very good from this team, I saw in many games how high our work rate is and that is often very good. Today I have some criticism. It is not only the team that has to look in the mirror, I am part of it.'
The under-pressure United boss will be hoping his team produce a markedly improved performance when they host Tottenham in Sunday's huge Premier League clash.