Worcester had taken the lead with a try from centre Sam Tuitupou and then a Matthew Jones penalty, before Wasps hit back through England winger Paul Sackey.
Dave Walder, making his return from a broken ankle, slotted a penalty to push Wasps ahead - but he missed a crucial late shot at goal.
Then nine minutes into injury time, Miles Benjamin caused panic in the Wasps defence - and Crichton was able to step up to seal the victory.
Worcester had had their chances before then, and it appeared their decision to spurn a shot at goal after hammering away at the Wasps line was going to cost them.
In the end, assistant coach Clive Griffiths was able to blow out his cheeks in relief at a thrilling finish.
Wasps describe this as the first year AD - after Dallaglio - and the so-called "new generation" still do not appear entirely comfortable without their talismanic leader.
They were comprehensively out-classed in last weekend's season-opening defeat to London Irish and were forced to battle hard again today on an attritional afternoon at Adams Park.
Not only did they have to settle for a losing bonus, but there are injury concerns over Simon Shaw, Phil Vickery and captain Raphael Ibanez.
Worcester, without new dad Matthew Powell and Achilles injury victim Rico Gear, took the lead after 20 minutes when Josh Lewsey mis-judged Jones' high-ball and spilled it deep inside his own 22.
Tuitupou, who had over-run the original kick, pounced on the loose ball ahead of a desperately retreating Wasps defence to dive over the line.
It was little more than the Warriors deserved after a bright opening, epitomised by the all-energy performance of open-side flanker James Collins.
The academy graduate injected a combative edge that was missing from the Warriors against Northampton last week, and his dynamic efforts in the loose kept Wasps rattled.
Walder was struggling to make his mark and missed a long-range penalty, before firing a drop-goal attempt wide.
Worcester took a quick 22, and that momentum carried them downfield - where the powerful Tuitupou scored near the posts.
Jones missed his conversion but slotted a penalty to extend the Worcester lead, before Wasps eventually found some fluency and forced the Warriors on to the back foot.
Some excellent continuity play carried Wasps into the Worcester 22 where they met some physical defence, not least from Tuitupou who floored James Haskell with a massive blindside tackle.
But scrum-half Eion Reddan created a gap with a nicely angled run from the back of a ruck and slipped a neat inside pass for Sackey to score under the posts and bring Wasps back to within a point at half-time.
Ibanez failed to return after the interval, and Worcester soon lost prop Chris Horsman just as they were defending a five-metre scrum under their own posts.
Worcester produced a huge scrum to win back possession, only to lose it again as Wasps laid siege to the Warriors' line.
Worcester held out valiantly as Sackey, Riki Flutey, Haskell and replacement hooker Rob Webber all drove for the line.
After hammering away, but failing to chisel a gap in the Worcester defence, Wasps earned a penalty and settled for the three points.
Wasps stepped up the intensity and pace of their rugby but could not find the cutting edge - and Worcester almost turned the tables.
Dale Rasmussen was held up over the line as Worcester built on Jones' remarkable 70-metre kick for touch and a stolen line-out from Webber's first throw.
Worcester hammered away at the Wasps line and earned a penalty. They decided against going for goal. It was a gamble and appeared to have cost them the game as Wasps forced the turnover and cleared their lines.
Worcester did think they had scored again when Latham ghosted over the line following a slick midfield move - but play was called back for a forward pass.
Walder missed a late penalty shot - and ultimately, that proved decisive. Benjamin attacked, and Crichton stepped up to win the match.