Nothing was certain until Lawrence Dallaglio emerged from a pile of bodies for a 72nd-minute try which took the match out of reach. Rudi Straeuli's visit to the sin-bin a minute later did not help the Bedford cause either, because Mike Friday was able to get in on the scoring act as well.
Kenny Logan finished with 20 points, all from the boot, making him the fourth player past 100 points this season.
But Bedford can hold up their heads, with the pack a personification of the whole team's commitment. If Tony Yapp's drop goal just 83 seconds into the match did not serve as a warning of Bedford's intentions Wasps only had to consider the way in which their opponents hurled their bodies into the fray. It was not until they had fallen further behind to Sam Howard's first two penalties that Wasps shook themselves awake.
The Ireland centre Rob Henderson took the ball on the burst, broke three tackles and, from the ensuing ruck flanker, Florent Rossigneux picked up and galloped between the posts. Logan's conversion and two penalties restored a more natural order to affairs by the interval.
The Scotland wing increased that lead early in the second half, punishing further offences by Bedford with his increasingly reliable boot. Straeuli won a line-out and flanker Jason Forster was driven over for a try, which Howard converted to level the scores, but after that Wasps finally broke clear.
Wasps: Tries Rossigneux, Dallaglio, Friday; Conversion Logan; Penalties Logan 6. Bedford: Try Forster; Conversion Howard; Penalties Howard 3; DG Yapp.
Wasps: J Lewsey; S Roiser, L Scrase, R Henderson, K Logan; A King, A Gomarsall (M Friday, 64); D Molloy, T Leota, W Green, M Weedon (capt), S Shaw (A Reed, 73; D Macer, 80), L Dallaglio, J Worsley, F Rossigneux.
Bedford: S Howard; R Underwood, J Ewens, B Whetstone, D O'Mahony; T Yapp, C Harrison; N Hatley, J Richards, C Boyd, D Zaltzman, S Murray (A Duke, 49), J Paramore, R Straeuli (capt), J Forster.
Referee: C Rees (London).