My non-combat resolution system works great, it does a wonderful job of keeping everyone at the table immersed in their characters and the fantasy world they are playing in. There is no faster way, however, to pull them out of that world than when weapons come out and combat ensues. Suddenly you've gone from deep role-play and vivid imagination to a series of bland dice rolls with lots of misses.
I'm currently running my low-level group through Keep On the Borderlands and they have just begun to explore the Caves of Chaos. The castellan told them of an Ogre harassing and killing his men along the roadway north and he needed someone to go deal with the problem by bringing back its head. While the group was buying supplies, the blacksmith's 16 year old apprentice, "Wort", a randomly rolled level 0 fighter-in-training (who happened to have extraordinary strength and constitution @ 18 & 18 respectively) decided to join up with them for the promise of adventure. So we have a Fey Elf, Demon Hunter (Cleric mod), and an iron-pumping teen fighter.
The group arrived at the caves in the evening and decided to setup camp in the grove just outside the Ogre's cave, which they just so happened to explore first the following morning. I've been trying hard to incorporate my non-combat system into combat as much as possible. I honestly don't ever want to fully remove combat rolling because I think you get into a place that no longer resembles D&D anymore and you've got to maintain some semblance of stat-based attack and defense. But, there is quite a bit of bending you can do to make things far more cinematic.
As soon as the Ogre was alerted to the group's entry into his cave, he came stomping out to see what was going on giving enough time for the group to realize that some huge "thing" was coming for them. Ogre's (and most of the early level creatures) are pretty stupid anyway compared to my PCs so it seems fair for the creature to forget any sort of intelligent ambush tactic. In many cases, I skip the surprise round because there is usually a party that is clearly being more careful and methodical than the other.
The elf immediately cast entangle causing the tree roots from above to grab hold of the Ogre's arms. The monster failed his saving throw, so he was unable to move his upper body for the rest of the round. He could still kick and hop though, so keeping your distance from his lower body became important. The demon hunter had found a Valerian dagger (dagger +1) during his last adventure (Valeria is an extinct kingdom of sea worshippers - think Atlantis) and wanting to use it badly, decided to climb the vines and swing onto the beasts shoulders. The demon hunter has very good dexterity, and since the Ogre couldn't move I arbitrarily assigned him a d100 70% chance of success, which he succeeded. In this position, the Ogre could not fight back against the attackers slashing at his neck and head.
The elf later did the exact same thing while the fighter apprentice slashed at the monsters knees with his longsword. The Ogre failed his save a second time and the fighter landed the killing blow with a d8 result of 7 + 3 (str bonus). The whole thing was very cinematic and took surprisingly few dice rolls to complete.
A few moments later, when the Ogre's goblin friends became alerted to what was going on in the room below them, a couple brave ones began climbing down a make-shift rope latter to attack the heroes. The players had the brilliant idea to have Wort shake the bottom of the ladder, throwing the creatures off and instantly killing them by getting dashed against the rocky cave walls. They then used their torch to burn the trapdoor, the smoke filling the goblin chamber above and causing the creatures to flee. All this without a single combat roll.
Visualizing the battlefield with all its nuances is key to running combat encounters this way. When you minimize the transition to and from combat, I think the suspension of disbelief can keep going strong in players' minds. A brilliant idea to kill or disable your enemy before the dice even start rolling I think is wholly in the spirit of classic D&D. The player characters were only level 2, 1 and 0, they really had no business taking on a level 4 Ogre along with 12 goblins at that stage of the game, but using their heads they were able to not only able to minimize the risk, but neutralize it entirely. Nobody lost a single hit point.
I used to run combat with bland dice rolls: "roll for initiative", "you missed, he missed, you hit, he missed, you hit, etc". I've learned along the way that it doesn't have to be like this at all, not even close.