Steve McLaren's men did most of the pressing throughout but were made to rue a great early opportunity fluffed by Luc Castaignos and could even have lost the game late on as Levante counter-attacked dangerously.
Angel pulled his shot wide when clean through with three minutes remaining before Nikolay Mihaylov produced a brilliant save to turn Jose Barkero's shot wide moments later.
The draw, allied to Hannover's 3-2 win over Helsingborg, means the German side have now qualified with Levante second on seven points, Twente on three and the Swedes in fourth with just a solitary point to their name.
Things could have looked a lot more positive for the Dutch side though had Cataignos been able to make a proper contact when he was presented with a free header just six yards out after six minutes.
Robbert Schilder was then denied by a fine save from Keylor Navas from point-blank range midway through the half, whilst Pape Diop had Levante's best effort of the opening period but his long-range effort was comfortably held by Mihaylov.
Navas was called into action again 10 minutes after the break to turn Roberto Rosales' effort wide before Douglas should have given the hosts the lead from a corner but he too completely scuffed his shot when presented with a clear sight of goal.
The game then drifted for a while but came to life again in the final five minutes as Navas produced another good save low to his right to parry a long-range effort from Edson Braafheid and seconds later Angel was running through on goal at the other end but pulled his shot wide when faced with Mihahlov.
Twente's Bulgarian keeper then made arguably the save of the evening to turn Barkero's effort behind from another counter but Levante were content to travel home with the point that sees them on the brink of qualification.