I would be surprised if Soyuz rocket will not win.
The exact statistics could not be available, I'm afraid, but Soyuz rockets made more than 900 successful launches (over 1800 if you count the whole R-7 family). Some info can be found here, but only since year 2000.
Payload mass differs according to orbit type, but to assume an average mass of about 4 tonnes (underestimation, I think) we obtain the estimation 900 x 4 tonnes = 3600 t; for all R-7 we get more than 7000 t.
The question is how to count Space Shuttle. If Orbiter is included, than estimation is above 135 launches x 100 tonnes = 13500 t - much more than Soyuz and more than all R-7. But if don't count Orbiter, than it's not more than 135 x 20 tonnes = 2700 t.
For other launch systems:
- not more than 84 launches x 20 tonnes (overestimation) = 1680 t
- about 103 launches x 10 tonnes = 1030 t (most launches are to GTO)
- rather small amount of the launches were to LEO. Most to GTO/interplanetary. To assume average payload mass ~7 t (rather fair, I think) estimation is 363 launches x 7 tonnes = 2541 t, still less than Soyuz.
Titan rocket family:
- hard to collect exact statistics, but according to Wikipedia had total 363 rockets built, including 107 of heavy configurations (12 of Titan 34D, 31 Titan IIIC, 22 Titan 3D, 6 Titan IIIE, 35 Titan IV). For heavies 107 * 15 tonnes (gross overestimate) = 1605 tonnes. Light configurations had no more than 3 tonnes LEO payload capacity, also most of them vere used for suborbital launches.
Chinese Long March -2, -3, -4:
- also can be considered as family, because they are technically similar. Can't find detailed statistics or even total number of launches. But we can surely assume the number of launches if far less compared to Soyuz, with payload similar to Soyuz or less (3-9 tonnes).